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Iran: the next generation
Reza Marashi and Jason Rezaian write:
For several weeks now, observers and analysts of Iran have been referring to an emerging rift between the Iranian president Mahmoud Ahmadinejad and the country's Supreme Leader, Ayatollah Ali Khamenei. The recent back-and-forth between Ahmadinejad and Khamenei reflects a deeper generational shift. After three decades of Iran's clerical network dominating the political scene, the emergence of the Islamic Republic's next generation of leaders—nonclerical, war-veteran technocrats—may well portend larger ramifications for Iran's inward and outward orientation.
The common narrative argues that all Iranian leaders—especially given the vetting system that one must go through to enter politics—are cut from the same cloth. "He is one of them," or "He is like all the rest," has increasingly become the mantra of a society and much of its Diaspora who have grown tired of decades of disappointment. At the moment, though, Ahmadinejad and his cronies have emerged as an unlikely group challenging the status quo in Iran; simply put, when looking at the trajectory of the Islamic Republic and what it has stood for since its inception in 1979, the current president and his cabinet have done more to shake the system to its core than any other group, including their reformist predecessors.
This should not be taken as an endorsement of Ahmadinejad, or a suggestion that he intends to dismantle the system—far from it. But continuing to push the boundaries of what is acceptable by the Islamic Republic's own standards is certainly a trend worth tracking. It is through this paradigm that the recent rift between Ahmadinejad and Khamenei tells the real story: It's not about Ahmadinejad as much as what and who he represents: a generation of war veterans who felt the Iranian power structure had cast them aside. This generation increasingly personifies everything that Iran's clerical establishment is not; they are seen as young and confident; as the real reason for Iran's revolutionary survival and at the heart of a dissipating mistrust of the West in the wake of the Iran-Iraq war. Above all, they represent a belief system predicated on Iranian self-reliance and self-sufficiency. They have remained loyal to the Supreme Leader for religious reasons, but are hostile towards clerics who grabbed power while they fought to protect Iran from Iraqi aggression. To that end, they believe that the Islamic Republic has become corrupt and deviated from the true path of the 1979 Revolution. Perhaps more than seeking to profit from their inclusion amongst Iran's political elite, this new generation of technocrats seeks to include Iran more fully in the global economy.
This entry was posted in Iran on July 1, 2011 by News Sources.
← Clashes ease in Cairo, but underline Egypts fragile condition Saudi Arabia worries about stability, security and Iran →


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Charles-Félix Cazeau (24 December 1807 – 26 February 1881) was a French Canadian priest and administrator of the Archdiocese of Quebec who was prominently involved in the relief of victims from the Great Irish Famine (1845-1849).

Cazeau began his classical education in 1819 at Quebec City. He studied at the Collège de Saint-Roch which had been recently founded by Bishop Joseph-Octave Plessis and one of his teachers was a future archbishop of the Archdiocese of Quebec, Charles-François Baillargeon.

References

External links
 the Catholic Encyclopedia

1807 births
1881 deaths
19th-century Canadian Roman Catholic priests
History of Catholicism in Quebec
People from Quebec City

redpajama_set_name: RedPajamaWikipedia


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Our Stories Archive
The First Day of School, But Now Youre the Teacher
Morrison "Mo" Robblee '19 will begin his teaching career this fall at Reid Middle School in Pittsfield, Mass.
Morrison "Mo" Robblee '19 has pictured himself as a teacher for a long time—and this fall, that picture will come into focus. Robblee, originally from Minnetonka, Minn., has accepted a full-time teaching position at Reid Middle School in Pittsfield.
After being recommended for a long-term substitute social studies teacher position at Reid by his student teaching program supervisor, Robblee applied for a permanent role in the school's social studies department.
"It feels incredible to be on the verge of accomplishing a major life goal," Robblee said. "I have been dreaming of this moment for the better part of a decade. Of course, there are feelings of nervousness and anxiety at the feat to be accomplished, but nothing can compete with the joy and excitement I feel to be where I am today."
Robblee used to visit family in Massachusetts as a kid, and knew he wanted to attend college in the Commonwealth. By the time he got to campus, he had already racked up college credits taking AP courses; because of this, he was able to finish his undergraduate work in three years, then shift into master's coursework.
"I've always wanted to work with people in a manner in which I could help them," he said. "I really like interacting with history and getting kids excited about history—so that is the path I chose for myself."
As a double major in history and education, he learned about educational theory, curriculum development, and lesson planning while taking courses on the history of women in the United States and the relationship of eastern and western European countries, among other things. MCLA's education program connects students with area teachers, and in addition to his long-term substitute job at Reid, Robblee had the opportunity to spend a full semester student-teaching at Hoosac Valley High School.
"I had full responsibility—you're designing lessons, you're implementing lessons, you're grading student work, you're assessing student achievement," he said. In order to get more comfortable in front of a class, he has also worked as a substitute teacher, camp counselor, and supplemental instructor. "All those experiences went really well and definitely affirmed that's the direction I'd like to go in," he said.
Robblee originally planned to continue to attend MCLA full-time, earning his master's in education through the College's 4+1 program. His new job has changed those plans: He'll continue working on his Me.D. and hopes to take at least one class a semester. (He got a jump-start by earning 12 credits toward his grad degree and teaching licensure this summer.)
School starts very soon—so Robblee is preparing, reaching out to his mentors and supporters, and getting excited.
"I have been reflecting upon my experiences as a student, asking for 'first-year teaching' advice to anyone I bump into, and expressing gratitude to the people who have helped guide me along this journey," he said. "I am most looking forward to getting to know all my students! We have got ourselves one interesting journey ahead us and, for all to be successful, I am looking forward to gaining the trust and respect of my fellow classroom adventurers."
Are you an MCLA student, alumni, or faculty member? Do you want MCLA to share your story? Please email Creative and Brand Strategy Manager Francesca Olsen at Francesca.Olsen@mcla.edu.

redpajama_set_name: RedPajamaCommonCrawl


REAL MADRID could put themselves on another collision course with Tottenham once again, as they are prepared to stake their interest in Paulinho.
The La Liga giants are ready to raid the north London club for a third successive year, after forking out a world-record £85 million to land Gareth Bale in the summer and £32m on playmaker Luka Modric the summer before that.
Paulinho, who only joined Spurs in a £17m deal in the summer turned in an excellent performance against Stoke, as Tim Sherwood's side cruised to a 3-0 victory.
However, Spurs may have to brace themselves for another unwanted bid from Real who have fixed their sights on Paulinho.
It is believed that Carlo Ancelotti has identified Paulinho as the type of player he wants to bring to the Bernebeu.
Although reports suggest Real would delay making a bid until the summer, Ancelotti is thought to be eager to launch a bid in the forthcoming window, with other European clubs harbouring Paulinho's signature.
Spurs are formulating their plans for what is gearing up to be a busy transfer window, with Erik Lamela linked to Atletico Madrid and Inter, while Sherwood may plump for Javier Hernandez, as the club faces the prospect of waving goodbye to Jermain Defoe.
Spurs moved to seventh courtesy of their win yesterday but face a stiff test of their European credentials when they face Manchester United on New Year's Day.


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Home News & Headlines US News Hurricane Hilary reaches Category 2 and propitiates storms in western Mexico
Hurricane Hilary reaches Category 2 and propitiates storms in western Mexico
Hurricane Hilary, which was formed today in Pacific waters, increased the strength of its winds and reached tonight category 2 on the Saffir-Simpson scale, reported the National Meteorological Service (NWS) today, and warned of storms in the states West and southwest Mexico.
The phenomenon, which travels west-northwest at 15 kilometers per hour, is tonight 470 kilometers south-southwest of Manzanillo, western state of Colima, and 980 kilometers south-southeast Cabo San Lucas, Baja California Sur (northwest).
Hilary develops sustained winds of 155 kilometers per hour and gusts up to 195 kilometers per hour, according to the SMN in its most recent report.
The institution indicated that the cloudless landslides favor the potential of very strong storms with intense points (75 to 150 millimeters) in Michoacán and very strong (50 to 75 millimeters) in Jalisco, Colima and Guerrero.
Likewise, he indicated, there is foreseen high swell of 2.5 to 3.5 meters on the coasts of Jalisco, Colima, Michoacán and Guerrero.
Therefore, the SMN recommended to the general population and to maritime navigation to take precautions for rain, wind and waves and to comply with the recommendations issued by the authorities of the National Civil Protection System in each entity.
According to the forecast, Hilary will be located Tuesday at 19.00 (2400 GMT) 560 kilometers southwest of Perula Beach, Jalisco, and 595 kilometers west-southwest of Manzanillo, Colima, as a category 3 hurricane.
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Rita B 'Remembering Patsy Cline'
Patsy Cline Tribute Acts
Versatile Solo/Duo Performance
Blue Moon Of Kentucky
Walking After Midnight
Rita B is a UK award winning country singer/songwriter. Rita has a wide range of music for everyone, from country to pop with some of her own work included. Rita performs an excellent tribute to Patsy Cline. Her all round performance and attitude towards her guests is first class. She is a very friendly person who puts her heart and soul into every song and in the words of Patsy Cline, "Oh, I just sing like I hurt inside" and that's exactly how Rita delivers. Rita is certainly an artist to look out for on the country music scene.
Accompanying Rita in her 'Remembering Patsy Cline' act is steel guitarist Micky Byrne. Micky has been a musician since he was 15 years old and took up the pedal steel guitar in 1970. Micky has backed some of the top Nashville names in the 80's on their tours of U.K. and Germany and he also toured the States in 1984. As a duo, their performance is outstanding.
Rita can also provide a trio, with lead guitarist Lee Hodgson who is the other half of Rita's duo, "Kentucky Cousins". Lee has played thousands of gigs, including tours with Number 1 pop act, Odyssey and jammed with Stevie Wonder's Wonderlove. Lee appeared with Memphis Roots at Wembley Arena and the Royal Albert Hall plus countless clubs, festivals etc. and continues to play UK country festivals and clubs with Boney Fingers, the Steve Cherelle Band and with Rita B as "Kentucky Cousins".
Back In Baby's Arms
A Church, A Courtroom, Then Goodbye
Crazy Arms
Eyes Of A Child
Faded Love
Have You Ever Been Lonely
Heartaches
Honky Tonk Merry Go Round
How Can I Face Tomorrow
I Cried All The Way To The Altar
I Don't Wanna
I Fall To Pieces
I Love You Honey
I'm Blue Again
If You've Got Leavin' On Your Mind
In Care Of The Blues
It Wasn't God Who Made Honky Tonk Angels
Just Out Of Reach
Lovesick Blues
Poor Man's Roses
Rose Of San Antone
Seven Lonely Days
Shake Rattle And Roll
She's Got You
So Wrong
Somday You'll Want Me
Stranger In My Arms
Tennesse Waltz
There He Goes
Three Cigarettes In An Ashtray
When I Get Through With You
Why Can't He Be You
Your Cheating Heart
All I Ever Need Is You
Banks Of The Ohio
Blacks Hills Of Dakota
Blanket On The Ground
Cherokee Boogie
Crying Time
Dance In My Fire
Dear Father
Do You Wanna Make Something Of It
Don't Be Stupid
Don't It Make My Brown Eyes Blue
Down At The Twist And Shout
First Cut Is The Deepest
Got Caught When Your Guard Was Down
Hard Way
He Thinks He'll Keep Her
Heads Carolina, Tails California
Heartbreak Usa
Here Comes My Baby
Here For The Party
Here You Come Again
Hey Good Lookin
Hometown Kentucky
Honey I'm Home
I Don't Feel Like Lovin You Again
I Want My Song To Have A Good Hook
It's A Heartache
Just Wanna Dance With You
Ka Ching
A Little Past Little Rock
Me Bobbe Mcgee
Mirror Mirror Why
Old Fashioned Song
One Bud Wiser
Our Shooting Stars
Our Song
Paper Roses
Pay That Bill
Redneck Woman
Rita's Song
Route 61 From Lousiana
Running Through My Veins
Send A Message To My Heart
Skin To Skin
Something In Red
Speak To The Sky
Takin In Easy
Tennessee Waltz
Then Love Was Young
There's A Guy Works Down The Chipshop
There's Your Trouble
This Is Not Imagination
Today I Started Loving You Again
Try A Little Tenderness
Unlove Me
Unloved Yearning
Vibrating Violins
Walk Right Back
Wasted Tears
When I Think About Cheatin
When Will I Be Loved
Whose Bed Have Your Boots Been Under
Why Have You Left The One You Left Me For
Will The Circle Be Unbroken
Wrapping Chords
You Always Hurt The One You Love
You're My Best Friend
Silver Pistols Country Music Club ("cmc")
Suffolkoutlaw Cmc
Coventry Hurst Green Cmc
Oxtedwest Moon Shiners Cmc
Chathampedlars Cmc
Norfolkwhite Buffalo Cmc
Bourne, Lincssilicon Valley Cmc Newbury
Berkshirebrentwood Theatre
Uk Country Radio Awards Ceremony
Etcbrighton Railway Clubmundesley Holiday Park
Norfolkdovercourt Holiday Park
Pitsea & Vange Wmc
West Ham Social Club
Writtle Sports & Social
Rafa Club
Withamraft Club Chelmsford
South Woodham Ferrers Yacht Club
Kings Residents Club
"'Country singer Rita B has won the UKCountryRadio Listeners' Club Award for 2011 in recognition of her song, Got Caught When Your Guard Was Down. Rita was presented with the award at UKCountryRadio Fourth Birthday Party on September 17th. Accepting the trophy from UK Country music legend, John Aston, rita said, 'I am absolutely gobsmacked at everything. Thank you to everyone who has voted for me and thank you to UKCountryRadio which is doing a great job in promoting UK artists'. Rita is the third artist to win the online radio station's annual award, following in the footsteps of Jonny Williams in 2010 and inaugural winners, Richard Plank & Guy Charles in 2009......'"
'We had a very entertaining night of country music from solo artist, Rita B. She has a wide range of music for everyone with some of her own work included. The highlight of the evening was when Rita performed an excellent tribute to Patsy Cline. Her all round performance and attitude towards the guests were first class. She is a very friendly person who puts her heart and soul into every song Rita started with country music at a very young age but didn't feel the opportunity was right to hit the big arena until a few years ago. Since then she has not looked back. She is certainly an artist to look out for'.
- White Buffalo CMC, Bourne - Cross Country Magazine Review (Sep, 2011)
Rita B 'Remembering Patsy Cline' Offers:


1st: Eaton flied out to left.
3rd: Eaton lined out to center.
6th: Eaton lined out to shortstop.
8th: Eaton sacrificed to catcher, Difo to second.
9th: Eaton struck out looking.
1st: Turner lined out to left.
3rd: Turner grounded out to second.
6th: Turner grounded out to pitcher.
8th: Turner singled to deep center, Difo to third.
1st: Harper struck out swinging.
4th: Harper homered to right center (403 feet).
6th: Harper flied out to right.
8th: Harper grounded into double play, shortstop to first, Turner out at second.
2nd: Rendon singled to right center.
4th: Rendon lined out to right.
9th: Rendon singled to right.
2nd: Soto sacrificed to catcher, Rendon to second.
4th: Soto grounded out to second.
7th: Soto homered to left (372 feet), Rendon scored.
2nd: Zimmerman doubled to deep left, Rendon scored.
4th: Zimmerman struck out swinging.
7th: Zimmerman struck out swinging.
9th: Adams popped out to second.
2nd: Murphy grounded out to shortstop, Zimmerman to third.
5th: Murphy flied out to left.
7th: Murphy popped out to shortstop.
9th: Murphy singled to center, Rendon scored, Soto to third.
2nd: Wieters flied out to center.
5th: Wieters grounded out to first.
7th: Wieters lined out to left.
9th: Wieters singled to right, Soto scored, Taylor to third, Wieters to second on throwing error by right fielder A. García.
3rd: Milone flied out to center.
5th: Milone struck out swinging.
9th: Difo grounded out to shortstop.
1st: Carpenter reached on bunt single to third.
4th: Carpenter grounded out to second, Wong to third.
6th: Carpenter lined into double play, third to first, Wong doubled off first.
8th: Carpenter homered to center (406 feet), Wong and Wisdom scored.
1st: Molina reached on infield single to third, Carpenter to second.
2nd: Molina grounded into double play, third to second to first, Carpenter out at second.
4th: Molina flied out to right.
8th: Molina lined out to second.
3rd: J. Martínez reached on infield single to second.
5th: J. Martínez singled to right center.
7th: J. Martínez singled to left.
8th: J. Martínez singled to right center.
1st: Ozuna singled to left, Carpenter scored, Molina to second.
3rd: Ozuna flied out to right, J. Martínez to second.
5th: Ozuna singled to right, J. Martínez to second.
7th: Ozuna grounded into fielder's choice to pitcher, J. Martínez out at second.
8th: Ozuna grounded out to third.
1st: DeJong flied out to left.
3rd: DeJong flied out to right, J. Martínez to third.
5th: DeJong flied out to left.
7th: DeJong flied out to center.
9th: DeJong homered to left (380 feet).
1st: Gyorko hit by pitch, Molina to third, Ozuna to second.
3rd: Gyorko grounded out to second.
5th: Gyorko singled to right, J. Martínez to third, Ozuna to second.
8th: Gyorko homered to left center (425 feet).
1st: Bader flied out to right.
4th: Bader doubled to center.
5th: Bader grounded into double play, pitcher to catcher to first, J. Martínez out at home.
2nd: Wong flied out to center.
4th: Wong singled to center, Bader to third, Wong to second advancing on throw.
8th: Wong singled to center.
2nd: Mikolas doubled to left.
4th: Mikolas hit sacrifice fly to center, Bader scored.
6th: Mikolas bunt popped out to first.
8th: Wisdom singled to right, Wong to second.
Paul DeJong handed the Washington Nationals a second straight walk-off loss, capping a back-and-forth finish with a game-ending solo homer in the ninth inning of the St. Louis Cardinals' 7-6 victory Monday night.


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Prince Austine, a lover of God, after the release of MY SHINNING LIGHT (not too long ago), is out again with a new one. IMELA is a song of gratitude to JEHOVAH OVERDO. God has been faithful to his word. Through the years, He has proven himself to be reliable and dependable. And all I want to say is THANK YOU.
Psalm 107: 31 oh that men will praise my God.


A Guy Walks into a Bar (with a Fake ID). And Becomes a Felon.
In Illinois it is a Class A misdemeanor punishable by up to one year in jail and fines up to $2,500 for lending your license or state ID to, and or using somebody else's license or state ID.


Did you know turtles are a protected species? Anyone caught touching, riding or disturbing a turtle will be subject to fines up to $5,000 and imprisoned for not more than 6 months.
There are seven species of sea turtles in the world, and three of these inhabit the waters of St. John. The two most common are the green and hawksbill turtles, while the leatherback is rarely seen. Sea turtles spend most of their lives in the water only coming ashore to nest. Turtles travel thousands of miles a year. Visit the Sea Turtle Conservancy web site to see where they go.
Trunk Bay beach is named after the leatherback turtle because they used to nest there in large numbers. But wait –they are called leatherbacks but the bay was named Trunk Bay. Why? Leatherbacks when on the ocean surface look like a steamer trunk floating so the Danes and locals called them trunk turtles, so the story goes.
Turtles are egg layers and depend on quiet, dark beaches to lay their eggs in camouflaged nests, called clutches. The eggs take between 55 to 70 days to hatch depending on sand temperature. It has been found that warmer nests produce more females while cooler nests produce more males. Hatchlings usually emerge from the nest all together, and make what appears to be a mad dash for the sea. They are extremely vulnerable at this time to predators on land and in the sea. They will become disoriented if there is artificial light in the area. Here is more information on the lifecycle of a turtle.
Photo © Caroline Rogers, all rights reserved.
The green sea turtle can reach almost 40 inches in carapace length (top shell), weigh up to 500 pounds and feeds on sea grasses. They reach sexual maturity between 20-50 years. The green is the largest of all the hard-shelled turtles. They typically nest between June and September. They do not nest on St. John. During the season they lay about five nests (clutches), one every two weeks, and it takes about two months for the eggs to hatch.
Hawksbill turtles are the smallest of the sea turtles that frequent our waters. A mature hawksbill will weigh between 100 - 150 pounds and measures about 25 inches in length. Female hawksbills return every two to three years to the beaches they were born on to lay their eggs. They lay three to five nests per season with about 130 eggs each. Hawksbill turtles range throughout the Caribbean, Indonesia, Mexico, and Australia. The hawksbill has a distinctive and beautiful shell (carapace). They were harvested almost to extinction for their shell, which was used to create jewelry, combs and brushes. Their diet includes sponges, other invertebrates, and algae. They can be found along rocky ledges and coral reefs where they seek shelter.
The leatherback is the largest of the sea turtles and gets its name from its leathery like dark grey or black shell with whitish spots. It is the only sea turtle to lack a hard shell. These pelagic (open ocean) turtles have seven distinct ridges running the length of its carapace (shell). The leatherback is typically 4 to 6 feet long and weighs 600 to 1100 pounds. The largest leatherback ever recorded was almost 10 feet long and weighed 2019 pounds. They nest every two to three years, four to seven times a season, and lay an average of only 80 eggs. The eggs take about 65 days to hatch. Leatherbacks feed on jellyfish.
The greatest threats to these turtles are commercial fishing and marine pollution. They get caught in the fishing nets of large fishing vessels. Their diet includes jelly fish and unfortunately they cannot tell the difference between jellyfish balloons or plastic bags.
How can you help protect sea turtles?
Become a Sea Turtle Beach Patrol Volunteer.
Never touch or chase them! By touching, chasing or riding them, visitors disturb their eating, resting and even nesting habits.
Don't litter. Remember some turtles mistake balloons, and plastic for food.
Don't leave deep ruts, holes or tracks in the beach. The baby sea turtles get stuck in them when they hatch attempt to reach the sea.
Prevent light from shinning on beaches.
Do not drive on beaches. You risk crushing eggs.
Never disturb feeding or nesting turtles.
Never buy or sell sea turtle products.
Check out the North Florida U. S. Fish & Wildlife page on how you can help protect sea turtles.


Red hdmi is my current setup. Blue is my theoretical setup.
Right now my PC has surround and so does my PS4, I just have to switch the input on the av receiver. Now that I use headphones for gaming, I don't need the surround sound. Currently when playing PS4, I can still use my other monitor for internet but there is no sound. Would this setup allow me to use my PS4 with my headphones for game sound and my other monitor work for internet with surround sound? If all is good, would the switcher lose any quality or fps compared to a straight connection?
My takai Woofer was producing very low sound,how to solve that?


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\section{Introduction}

We consider the problem of streaming a file divided into small units called ``chunks'' over multiple links.  Each chunk may be requested on any link. Requested chunks arrive in the requested order, separated by random delays. Randomness is due to propagation delays, change in the link quality and congestion since the streaming flow shares the links with competing flows. Received chunks are stored in a buffer. After a fixed start-up time called pre-buffering time, the received chunks are played at a fixed speed. If a chunk does not arrive before the time it is supposed to be played there is starvation. For video streaming, starvation causes interruptions in the play-out and is the main cause for poor Quality-of-Service (QoS). The goal is to design chunk request policies that minimize the starvation probability.

For clarity we use the generic term ``links'' to denote entities on which packets may be requested. Links may represent for instance: (a) several disjoint paths between a source and a destination in a wired network (b) several disjoint frequency bands in a wireless system featuring spectrum aggregation, (c) several wireless interfaces which employ different protocols, for instance cellular data, say LTE (Long Term Evolution) and local area, say 802.11 (WiFi). Case (c) is often referred to as ``multihoming''. While links may have different data rates as well as different traffic management rules, a good protocol should ensure that, from the point of view of the streaming flow, they appear as a single aggregated entity whose data rate is the sum of the links data rates. The goal is hence to perform link aggregation efficiently so as to minimize the starvation probability.

The problem is motivated by several features of current networks. First, in current mobile networks, video-on-demand services consume the majority of the available bandwidth, and good traffic management rules have a dramatic impact on the network performance. Further, due to the massive deployment of both cellular data networks and local area networks, most users in dense urban zones have access to several wireless links. If feasible, spreading the load evenly on those links is highly desirable. Finally current streaming protocols such as the DASH standard~\cite{Stockhammer11} operate by splitting video files in chunks and requesting them as independent HTTP requests. There are two advantages to this approach: (i) this is a stateless protocol, the video server does not keep track of the state of the user playing the video, and processes each chunk request independently (ii) requesting each chunk through HTTP avoids issues with middle-boxes (e.g. firewalls and NATs) which may block certain ports for security.

There are two challenges. First the number of request policies is exponential in the number of chunks, so that exhaustive search is not feasible. Furthermore, one would like to minimize the amount of feedback required by policies to ease implementation. The second challenge is to  calculate the starvation probability of a given policy by tractable formulas. This enables to select the pre-buffering time to ensure that the starvation probability is smaller than some threshold. For elastic traffic one is concerned with the maximum achievable long-term throughput, which depends mostly on the expected delay but not on the full delay distribution. On the other hand for streaming traffic, starvation is highly dependent on the variability of the delays (e.g. the jitter), and accurate predictions of starvation probability must depend on the full delay distribution. Analytical formulas are useful even when one can simulate the system since, when the pre-buffering time is large, starvation tends to be a rare, ``large-deviations'' type of event. So one needs long simulations to get accurate estimates.

{\bf Our contribution}: We first derive lower bounds on the starvation probability of any policy. Those bounds enable us to derive the scaling of the pre-buffering time required to reach a target starvation probability. We further propose simple, implementable policies (without feedback) to determine how chunks may be requested on the various links. We derive upper bounds and/or approximations for the starvation probability of these policies. Those upper bounds are tractable, including both cases where delays are independent and correlated. We deduce that the proposed  policies are efficient since they approach the fundamental limits derived earlier. Finally we specialize our results to several situations of practical interest: (i) links that employ opportunistic scheduling or random access at the packet level (ii) links where the streaming flow is the secondary user of a cognitive radio network, and is allowed to use the link only when the primary user is inactive (iii) links where the streaming flow shares the available bandwidth with short-lived flows using (for instance) fair rate sharing.

The remainder of this article is organized as follows. In section~\ref{sec:related_work} we highlight the relationship between our results and previous work. In section~\ref{sec:model} we state our assumptions and notation.  We provide performance limits and state the proposed schemes in section~\ref{sec:limit_algo}. In section~\ref{sec:performance} we calculate the performance of the proposed schemes under general assumptions. We specialize our results to several popular link models in  section~\ref{sec:case_studies}. In section~\ref{sec:numerical} we illustrate our results through numerical experiments. Section~\ref{sec:conclusion} concludes the article.

\section{Related work}\label{sec:related_work}
\subsection{Performance evaluation and control for streaming}

Streaming over wireless networks has become popular, and a lot of attention has been dedicated to its performance evaluation, to understand the relationship between pre-buffering and starvation. \cite{TMM10:Luan} considers a G/G/1 queue where the first two moments of the arrival and service distributions are known. The discrete buffer size is approximated by a Wiener process. \cite{TMM08:Liang} considers an ON-OFF  wireless channel and performs a probabilistic analysis based on the knowledge of the playback and arrival curves. An M/M/1 queue is used in \cite{Infocom12:Xu} to derive explicit formulas for the number of starvations, based on the Ballot theorem~\cite{Takacs}. \cite{YuedongInfocom13} considers the combination of fading and flow level dynamics, and calculates the starvation probability by solving a set of ordinary differential equations constructed on top of the Markov chain describing the flow level dynamics.

All these works considered a single wireless link in a classical TCP/IP architecture. Another set of works studied the impact of multihoming on streaming QoS. For instance, \cite{habib} studies multihomed streaming in a residential context using a DSL and a cable connection. Experiments show that connection splitting/migration in case of congestion provides a significant QoS improvement. For lossy wireless channels, \cite{JurcaRateAllocation} considers the rate allocation problem as an optimization problem to minimize a distortion metric for the video taking into account the loss rate of each of the network paths. \cite{JurcaFEC} extends this work to joint scheduling and Forward Error Correction (FEC), while \cite{multihome_WiFi_UMTS} considers the specific case of WiFi/UMTS multihoming in the design of the FEC scheme. The same objective of reducing the video distortion is used in \cite{multihomed_TON} to design a multipath scheduling policy. These works aim at reducing video distortion, while we are interested in HTTP-like video streaming with pre-buffering where video chunks are downloaded without distortion and the QoS metric is the starvation probability.

Another set of works on multihomed streaming is related to content centric networking where the rate allocation between paths is driven by interest packets generated by the users. Authors in~\cite{salsano_streaming} consider a group of neighboring users using their mobile devices to view the same video, and exploit the fact that users can exchange video chunks using direct connections instead of downloading all the chunks using the cellular link. Others consider {\em real time} services with stringent delay constraints for instance voice \cite{Jacobson_voice} and live TV \cite{Xu_TV}, which is out of the scope of the present work.

To the best of our knowledge, the present work is the first to derive optimal allocation strategies for streaming over multi-homed access with the starvation probability as the main performance indicator and provide strong theoretical performance guarantees given in the form of tractable, analytical formulas.

Two contributions on multipath streaming which are complementary to ours are \cite{Singh13} and \cite{Xing14}. In \cite{Singh13}, the authors propose a protocol for multipath streaming and evaluate its performance experimentally but do not conduct a theoretical analysis. The authors of \cite{Xing14} formulate the problem as a Markov Decision Process (MDP). Since computing the optimal policy of this MDP is prohibitively expensive they propose a heuristic policy search algorithm for which no theoretical guarantees are available and its performance is evaluated numerically.

\subsection{Receiver driven transport protocols}
While IP transport is under destination control, several receiver-driven transport protocols have been proposed (see Gupta et al. in \cite{gupta1998webtp}). Receiver driven protocols have recently attracted a lot of attention with the development of Information-Centric Netwroking (ICN), where download rates are determined by the rate at which interests are sent into the network by the end-system. Arianfar et al. \cite{arianfar2010contug} discuss the challenges faced when designing transport protocols for content centric networks and present a receiver-driven transport protocol, ConTug, that takes into account the presence of multiple sources. Carofiglio et al. propose another receiver-based protocol called ICP (Interest Control Protocol) using AIMD window-based congestion control \cite{Carofiglio2012a,Carofiglio2013b}. \cite{Saino2013} proposes CCTCP (Content Centric TCP) that considers the use of multiple sources in the design of the protocol parameters such as the timeout timer.

In our work, we are neither interested in the design of such transport protocols, nor in the evaluation of their performance for streaming traffic. Our objective is to design optimal policies for splitting interests over the different interfaces. We nevertheless assume that mechanisms that adapt the interest sending rates to the state of each multihoming path are implemented, ensuring that the available capacity of each link is fully used. Our policies can thus be implemented at the upper layers over classical TCP/IP based protocols, or in the form of a receiver-driven interest control protocol.

\subsection{Spectrum Aggregation}
Spectrum aggregation is driven by two different but complementary objectives: achieving higher user data rates and exploiting all the available spectrum to offload traffic from the mobile network. Typical solutions to attain the former objective are the so-called carrier aggregation features in 3GPP, starting with the dual cell feature in 3G systems (Release 8). The dual cell feature aggregates two 3G carriers of 5 MHz and doubles the achievable data rate~\cite{Morais09}. However, the dual cell feature has been shown (\cite{Bonald09}) to achieve significant capacity gains for streaming services due to the trunking effect, especially since the data rate in Release 7 is not sufficient to meet the requirements of TV streaming . It is noted that this gain is reduced when MIMO is combined with the dual cell feature in Release 9. The term ``dual-cell'' has been changed to ``carrier aggregation'' in recent standards as more than two carriers are aggregated starting from Release 10. 3GPP then introduced a more advanced carrier aggregation mechanism, where the unused spectrum in the 1.4 GHz band (originally allocated for TDD mode but not used in Europe) is aggregated with the FDD LTE spectrum via multihoming \cite{3GPP}.

Another set of spectrum aggregation technologies aims at increasing the system capacity and offloading the cellular traffic to other systems, while offering a better QoS to users. This started with Cognitive Radio (CR) which suggests to reuse unused spectrum allocated to other services in a secondary, non-priority, way \cite{Mitola99}. The first commercial services related to CR reuse white spaces left by TV systems \cite{Nekovee09}. However, as spectrum availability cannot be guaranteed to secondary users, multihoming is advocated: users simultaneously connect to both licensed and non-licensed spectrum. 3GPP systems adopt multihoming to aggregate 4G spectrum with WiFi spectrum, in the so-called dual connectivity mode \cite{Zakrzewska13}, exploiting the presence of both WiFi and 4G radio frequency modules in almost all user devices. The 3GPP LWIP Release 13 standard includes aggregation of WiFi and LTE (called ``Wifi Boost'') at the IP layer, and \cite{Ling15,Perez16} show that large performance gains result from this aggregation.

As opposed to classical carrier aggregation where resource allocation is fully centralized,  systems based on CR and dual connectivity can be user-centric: users sense the medium and decide which packets to send to each carrier, possibly assisted by the network \cite{Dandachi16}. Decisions are taken by users based on local information (sensing) and contextual information (link quality estimates) provided by the network. Different carriers are usually heterogeneous in terms of radio channel quality (independent fast fading), coverage (path loss and slow fading) and load (number of connected users). The decisions taken by users have a large impact on QoS and an efficient protocol is needed.

\section{The model}\label{sec:model}

\subsection{Basic Description}

We consider a file divided in $N$ chunks of unit size, indexed by $n \in [\![ 1,N ]\!]$. There are $K \geq 1$ links, on which any chunk can be requested. When a requested chunk is received on a link, it is placed in a buffer. After a pre-buffering time denoted by $B > 0$ the file is read at unit speed. Namely, at time $n + B$, if chunk $n$ is present in the buffer then it is read, and otherwise starvation occurs.

We define the chunk request policy which is represented by a vector $\pi \in [\![ 1,K ]\!]^N$, where $\pi_n = k$ if chunk $n$ is requested on link $k$. We assume that if two chunks $n < n'$ are requested on the same link then $n$ is requested before $n'$. It is clear that this always ensures lower starvation. We define $d_k(n) = \sum_{n'=1}^{n} {\bf 1}\{ \pi_{n'} = k \}$, the number of chunks comprised between $1$ and $n$ requested on link $k$. We further define $X_k(\ell)$ the delay of the $\ell$-th chunk requested on link $k$. Namely, if $\pi_n = k$, chunk $n$ arrives at time $\sum_{\ell = 1}^{d_k(n)} X_k(\ell)$. The starvation probability $P^N$ is the probability that there exists a chunk that arrives after the time it was supposed to be played, so that:
\eqs{
P^N = \mathbb{P}\left[ \exists k, n : \pi_n = k \text{ and } \sum_{\ell=1}^{d_k(n)} X_{k}(\ell) > n + B\right].
}
It is noted that in general $\pi$ can be sample path dependent, so that $\pi$ depends on the delays $( X_k(\ell) )_{\ell,k}$. We will consider two types of policies.
 \vspace{-0.15cm} \begin{itemize} } %\setlength\itemsep{-0.1cm}
\item \underline{Static policies}: $\pi$ does not depend on $( X_k(\ell) )_{\ell,k}$
\item \underline{Oracle policies}: $\pi$ is an arbitrary function of $( X_k(\ell) )_{\ell,k}$
 \end{itemize}
Static policies may depend on the delay statistics but not on their realization, so that they require no feedback and are attractive in terms of implementation. Oracle policies are policies that ``know everything'', including the delays of chunks not yet received. Oracle policies are not implementable and mainly serve here as a performance upper bound.
\subsection{Assumptions}
We introduce several sets of assumptions on the delays.
\begin{assumption}[I.I.D. delays] \label{as:iid}
For all $k$, $(X_k(\ell))_{\ell \geq 0}$ is an i.i.d sequence with expectation $\mu_k$, variance $\sigma_k^2$ and cumulant generating function $G_k(a) = \log( \mathbb{E}[e^{a X_k(\ell)}] )$. We assume that $G_k(a) < +\infty$ on an open neighbourhood of $0$.
\end{assumption}
\begin{assumption}[Markovian delays] \label{as:markov}
	For all $k$ there exists a continuous time, stationary ergodic Markov chain on a discrete space ${\cal S}$ denoted by $( S_k(t) )_{t \in \mathbb{R}}$ with stationary distribution $m_k$ and transition rate matrix $Q^k = (q^k(i,j))_{i,j \in {\cal S}}$. There exists a function $r: {\cal S} \to \mathbb{R}^+$ such that for all $\ell,k$:
\als{
\tau_k(\ell) &= \min \left\{  t \geq 0: \int_{0}^{t} r(S_k(u)) du \geq \ell \right\}, \sk
 X_k(\ell) &= \tau_k(\ell) - \tau_k(\ell-1).
}
We define $\mu_k = \mathbb{E}[X_k(\ell)]$.
\end{assumption}
The Markovian case can be understood as follows: at time $u$, link $k$ is in state $S_k(u)$ and the streaming flow has an instantaneous data rate of $r(S_k(u))$. So  $\tau_k(\ell)$ is the duration required to receive $\ell$ chunks of unit size. Hence $X_k(\ell) = \tau_k(\ell) - \tau_k(\ell-1)$. The state is typically the number and/or state of the other flows sharing link $k$ with the streaming flow. For both sets of assumptions we define $r_k = 1/\mu_k$, the average data rate of link $k$, and $R = \sum_{k=1}^K r_k$ the sum of data rates. Based on the value of $R$ we distinguish three regimes: underload if $R > 1$, critical if $R = 1$ and overload if $R < 1$. We define the frequency vector $f = (f_1,...,f_K)$, with $f_k = r_k/R$. In order to balance the loads of the available links, the number of chunks requested on link $k$ should be close to $N f_k$, hence the name "frequency" for $f_k$. We shall see that the proposed algorithms precisely accomplish that goal. We denote by $\Psi(x) = \frac{1}{\sqrt{2 \pi}} \int_{x}^{+\infty} e^{-\frac{z^2}{2}} dz$ the complementary c.d.f of the standard Normal distribution. For clarity, we denote by $P^N(\pi,B)$ the starvation probability for $N$ chunks, prebuffering time $B$ and policy $\pi$. Finally we introduce an assumption on the independence of delays across links.
\begin{assumption}[Independent Links] \label{as:links}
$(X_k(\ell))_{\ell \geq 0}$ is independent from $(X_{k'}(\ell))_{\ell \geq 0}$ for all $k \ne k'$.
\end{assumption}
\subsection{Applicability of the model and rate adaptation}
Two remarks about the model are in order. First it should be noted that the Markovian delay assumption includes \emph{any} delay distribution and \emph{any} form of correlation between the successive delays observed on a given link. Indeed, there are no restrictions on the size of the state space $\cal S$ nor on the transition rates $Q^k$. For instance the process $(S_k(t))_k$ can follow any of the classical queues (as considered in prior work) say M/M/1, G/G/1 etc.

Second, throughout the article we assume that the video data rate is fixed and it is taken to be unity in our notation simply to ease presentation. Indeed, in practice, some form of rate adaptation will be implemented: before the video is played, the data rates $r_1,...,r_K$ will be estimated, and the data rate of the video will be chosen approximately equal to $\sum_{k=1}^K r_k$. This guarantees that the video quality is as high as possible while starvation probability can be made small by choosing the pre-buffering time appropriately and using an efficient policy to request chunks.

Performing rate adaptation is relatively simple since it involves only first order statistics, namely the first moments of the delays $\mu_k$, since $r_k = 1/\mu_k$. On the other hand selecting the pre-buffering time to avoid starvation seems more challenging since it involves the complete distribution of delays including higher moments (e.g. the jitter), as starvation is mostly a "tail event" due to large, rare fluctuations of the delays.

\section{Performance limits and algorithms}\label{sec:limit_algo}
\subsection{Fundamental Limits}
 Our first result presented in theorem~\ref{th:iid_lower_bound} is a lower bound on the starvation probability that holds for all oracle policies (hence for static policies as well). The intuition behind statement (i) is simple: if starvation does not occur, then, for all $n$ the total number of chunks received on all links at time $n + B$ must be greater or equal to $n$, otherwise there exists a chunk $n'$ which has not been received at time $n' + B \leq n + B$. The lower bound is not explicit but may be computed easily by simulation. Statement (ii) shows that for large files ($N \to \infty$), if $R \leq 1$, then the pre-buffering time must be of the order $(R^{-1} - 1)N + {\cal O}(\sqrt{N})$, otherwise the starvation probability tends to $1$.  For large files, we obtain an explicit lower bound using the central limit theorem. This lower bound is an increasing function of the delay variances $\sigma_k^2$, and tends to $1$ when we let $\min_k \sigma_k^2 \to \infty$. As a consequence, in the critical case, if the delays are heavy tailed, one cannot ensure a starvation probability $< 1$ with ${\cal O}(\sqrt{N})$ pre-buffering time. Also there are sharp transitions between the regimes: from $B = {\cal O}(1)$ (underload) to $B = {\cal O}(\sqrt{N})$ (critical) to $B = {\cal O}(N)$ (overload).  We will subsequently show that these orders are tight. Also note that, intuitively, the law of iterated logarithm would suggest the pre-buffering time to be at least ${\cal O}(\sqrt{N \log\log(N)})$ in the critical regime and we show that this intuition is flawed.

\begin{theorem}\label{th:iid_lower_bound}
	The following holds for all oracle policies $\pi$.

	(i) For all $B \geq 0$ and $N \geq 0$ we have:
	\als{
	P^N(\pi,B) &\geq \mathbb{P} \left[ \exists n \in [\![ 1,N ]\!] : \sum_{k=1}^K D_k(n) < n  \right] \sk
	D_k(n) &= \max\{ d \geq 0: \sum_{\ell=1}^{d} X_k(\ell) \leq B + n\}.
	}
	(ii) Let assumptions~\ref{as:iid} and \ref{as:links} hold. If $R \leq 1$, for all $b \geq 0$:
	\als{
		\lim \inf_{N \to \infty} P^N(\pi, (R^{-1} - 1)N + b\sqrt{N} ) &\geq \prod_{k=1}^{K} \Psi\left( \frac{b}{ \sigma_k \sqrt{ f_k }} \right).
	}
\end{theorem}

\noindent{\textbf{Proof.}}\
(i) Consider a fixed oracle policy $\pi$. Define the event
\eqs{{\cal A} = \left\{ \exists n \in [\![ 1,N ]\!] : \sum_{k=1}^K D_k(n) < n  \right\}.
}
Consider $n$ fixed and assume that $\sum_{k=1}^K D_k(n) > n$. Define $N^{\pi}_k(n)$ the number of chunks received on link $k$ before time $B + n$, and $N^{\pi}(n) = \sum_{k=1}^K N_k^{\pi}(n)$ the total number of chunks received before time $B + n$. By definition $N^{\pi}_k(n) \leq D_k(n)$ for all $k$, so that $N^{\pi}(n) \leq \sum_{k=1}^K D_k(n) < n$. Hence, at time $n + B$, there exists a chunk $n' \leq n$ which has not been received. We deduce that if ${\cal A}$ occurs, starvation occurs which proves the first claim.

(ii) Define $B^N =  (R^{-1} - 1)N + b\sqrt{N}$. Define the following events:
\eqs{
{\cal B}^N_k = \{ D_k(N) < N f_k \} \;\;,\;\; {\cal B}^N = \cap_{k=1}^K {\cal B}_k.
} If ${\cal B}^N$ occurs, $\sum_{k=1}^K D_k(N) < N \sum_{k=1}^K f_k = N$  so that starvation occurs.  Event ${\cal B}^N_k$ occurs if and only if \eqs{
\sum_{\ell=1}^{N f_k} X_k(\ell) > N + B^N,
}
so that, replacing $B^N$ by its expression:
\eqs{
	\frac{1}{\sqrt{N f_k}} \sum_{\ell=1}^{N f_k} (X_k(\ell) - \mu_k) > \frac{N + B^N - N f_k \mu_k}{\sqrt{N f_k}} = \frac{b}{\sqrt{f_k}}.
}
By the central limit theorem:
\eqs{
\frac{1}{\sqrt{N f_k}} \sum_{\ell=1}^{N f_k} (X_k(\ell) - \mu_k) \tto{N \to \infty} {\cal N}(0 , \sigma_k^2),
}
in distribution so that:
\eqs{
	\mathbb{P}[{\cal B}_k^N] \tto{N \to \infty} \Psi\left( \frac{b}{ \sigma_k \sqrt{ f_k }} \right).
}
Events ${\cal B}_1^N,...,{\cal B}_K^N$ are independent so that
\eqs{
P^N(\pi, B^N) \geq \prod_{k=1}^K \mathbb{P}[{\cal B}^N_k] \tto{N \to \infty} \prod_{k=1}^K \Psi\left( \frac{b}{ \sigma_k \sqrt{ f_k }} \right).
}
which gives the announced result.
\hfill $\Box$

\subsection{Efficient Algorithms}
Intuitively, to obtain an efficient policy, chunks should be requested on link $k$ at frequency $f_k$, so that the number of requested chunks $d_k(n)$ is close to $n f_k$. We define a class of static policies called upper balanced policies. We say that $\pi$ is $f$-upper balanced if for all $k, n$ we have:
\eqs{
	d_k^\pi(n) \leq (n + K - 1) f_k .
}
If frequencies are not rational numbers it is not obvious how to build such a policy. We show a simple recursive way to construct upper balanced policies with arbitrary frequencies.
\begin{proposition}\label{prop:up}
	Consider $\pi$ such that for all $n \geq 0$:
\eqs{
	\pi_{n} \in \arg \min_{k} \frac{d_k^\pi(n-1) + 1}{f_k}
}
with ties broken arbitrarily. Then $\pi$ is $f$-upper balanced.
\end{proposition}
\noindent{\textbf{Proof.}}\
	We drop ${.}^\pi$ for convenience. Define $a(n) = \frac{d_{\pi_{n}}(n-1) + 1}{f_{\pi_{n}}}$. Since $\pi_{n} \in \arg \min_{k} \frac{d_k^\pi(n-1) + 1}{f_k}$, we have  $a(n) \leq \frac{d_{k}(n-1) + 1}{f_{k}}$ for all $k$. Hence $f_k a(n) \leq d_{k}(n-1) + 1 $ for all $k$, and summing over $k$ we get:
$a(n) = a(n) \sum_{k} f_k \leq \sum_{k} (d_{k}(n-1) + 1)  = n - 1 + K$.

 Let us now prove that $d_k(n) \leq  (n + K - 1) f_k$ for all $n$. We proceed by induction. The inequality is true for $n = 0$ since $d_k(0) = 0 \leq (K - 1)f_k $.  Consider $n$ such that $d_k(n-1) \leq (n - 1 + K - 1)f_k$ for all $k$. If $k \neq \pi_n$ then $d_k(n) = d_k(n-1) \leq (n - 1 + K - 1)f_k \leq (n + K - 1)f_k$. On the other hand, if $k = \pi_n$ we have $d_k(n) =  d_k(n-1) + 1 = f_k a(n) \leq ( n + K - 1) f_k$. Therefore $d_k(n) \leq  (n + K - 1) f_k$ for all $n$ and all $k$ as announced.
\hfill $\Box$

Upper balanced sequences share a close resemblance with balanced sequences used in optimal routing problems such as \cite{Hajek85}.
A key difference is that it is usually difficult to compute balanced sequences when either $K \geq 3$ or frequencies are not rational. When frequencies are not rational, one cannot construct upper balanced sequences that are periodic, so that a (simple) recursive formula to calculate upper balanced sequences seems good enough. It is also noted that, contrary to problems of maximal throughput scheduling for elastic traffic, Bernoulli routing is not satisfactory. Namely using randomized sequences $\pi$ where $(\pi_n)_n$ is an i.i.d sequence on $\{1,...,K\}$ with distribution $(f_1,...,f_K)$ is no good since this would give $d_k^\pi(n) - n f_k = {\cal O}(\sqrt{n})$ (central limit theorem).

The policy defined in proposition~\ref{prop:up} can be implemented efficiently in practice as follows. As said previously, rate adaptation is performed so that $r_1,...,r_K$ and $R$ are estimated before chunks are requested. This allows to estimate frequencies $f_1,...,f_K$, and in turn calculate $\pi_n$ (the link on which chunk $n$ is requested) for all $n$ using proposition~\ref{prop:up}. This requires ${\cal O}(N)$ operations and ${\cal O}(K)$ memory.   An advantage of this policy is that it is completely static, hence it does not require to monitor the state of the various links once the streaming flow has started.

For very long files, the data rates of the various links may change on a slow time scale (say a few minutes), and to deal with this one may simply perform rate adaptation periodically and recompute $\pi$ based on proposition~\ref{prop:up}.

\section{Performance evaluation}\label{sec:performance}
\subsection{Performance for i.i.d. delays}
We consider i.i.d. delays. Our second result is theorem~\ref{th:iid_upper}, which gives upper bounds for the starvation probability of upper balanced policies, and shows that, in the 3 regimes of interest they are order optimal. Namely the pre-buffering times have the same scalings as the lower bound of theorem~\ref{th:iid_lower_bound}. More precisely, for statement (iii), we consider the regime where $R \le 1$. From our previous result the pre-buffering time must be greater than $(R^{-1} - 1) N  + {\cal O}(\sqrt{N})$, otherwise $P \to 1$ when $N \to \infty$. So we consider pre-buffering time $(R^{-1} - 1) N  + b + K-1$ and study the starvation probability as a function of $b$. Indeed, $P \not\to 1$ if and only if $b$ scales as ${\cal O} (\sqrt{N})$ when $N \to \infty$.

The proof is based on Doob's maximal inequality and before stating theorem~\ref{th:iid_upper} we define the exponent $a_k^\star$ which appears in our upper bounds. We define $F_k(a) = G_k(a) - a/f_k$. If $X_k(\ell) < 1/f_k$ a.s. then define $a_k^\star = +\infty$, and otherwise define $a_k^\star = \max \{ a \geq 0: F_k(a) = 0 \}$. Given $G_k(.)$, $a_k^\star$ can always be calculated numerically using a zero-finding procedure such as bisection or Newton's method. We further give explicit formulas for calculating $a_k^\star$ in several cases of interest: exponential delays and sub-Gaussian delays, see Proposition~\ref{prop:subgaussian}. We recall that $X_k(\ell)$ is sub-Gaussian if there exists $v^2_k \geq 0$ such that for all $a \geq 0$: $G_k(a) \leq a\mu_k + a^2 v_k^2/2$. It is noted that if delays are  Gaussian or bounded, then they are sub-Gaussian, see Remark~\ref{rem:subgaussian}.
\begin{proposition}\label{prop:subgaussian}
Consider $R > 1$.

(i) If $X_k(\ell) \sim$ Exp$(r_k)$, then $a^\star_k = r_k(1 + W( -R e^{-R})/R)$ with $W$ the Lambert function.

(ii) If $X_k(\ell)$ is $v^2_k$-sub-Gaussian then $a^\star_k \geq 2 \mu_k (R-1) / v^2_k$.
\end{proposition}
\noindent{\textbf{Proof.}}\
	(i) If $X_k(\ell) \sim$ Exp$(r_k)$, then $G_k(a) = r_k/(r_k - a)$, $a \in [0,r_k)$. So $a^\star_k$ is a solution to the equation:
	\eqs{\frac{r_k}{r_k - a} = \exp( a/f_k)}
Consider $v \in [0,R]$ and define $a=r_k(1 - v/R)$. Substituting in the equation above we get: $R e^{-R} = v e^{-v}$. Since $v \mapsto  v e^{-v}$ is strictly increasing on $[0,1]$ and strictly decreasing on $[1,+\infty)$ this equation has at most two solutions. $v = R$ is a trivial solution, which gives $a = 0$. The other solution is found by noticing that $-R e^{-R} = -v e^{-v}$, so that $-v = W( -R e^{-R})$, and $a =  r_k(1 + W( -R e^{-R})/R)$ which proves the result.

	(ii) If $X_k(\ell) \leq 1/f_k$ a.s. then $a^\star_k = +\infty$ so that the claim is trivially true. If we do not have $X_k(\ell) \leq 1/f_k$ a.s., then $F_k(a) \tto{a \to \infty} \infty$. Define $\overline{a} = 2 \mu_k (R-1) / v_k^2$. By continuity of $F_k$, to prove that  $a^\star_k \geq \overline{a}$ it suffices to prove that $ F_k( \overline{a}) \leq 0$. If $X_k(\ell)$ is $v^2_k$-sub-Gaussian, we have $$F_k(a) \leq a \mu_k  +  \frac{a^2 v^2_k}{2} - \frac{a}{f_k} = a \mu_k(1 - R) + \frac{a^2 v^2_k}{2}.$$ Setting $a \equiv \overline{a}$ we obtain that $F_k(\overline{a}) \leq 0$ which concludes the proof. \hfill $\Box$

\begin{remark}\label{rem:subgaussian}
	(a) If $X_k(\ell) \sim {\cal N}(\mu_k,\sigma^2_k)$ then $G_k(a) = a \mu_j + a^2 \sigma_k^2/2$ and $X_k(\ell)$ is $\sigma_k^2$ sub-Gaussian.

	(b) If $X_k(\ell) \in [\underline{x},\overline{x}]$ a.s. then $X_k(\ell)$ is $(\overline{x} - \underline{x} )^2/4$ sub-Gaussian (by Hoeffding's lemma).
\end{remark}

\begin{theorem}\label{th:iid_upper}
	Consider $\pi$ an $f$-upper balanced policy. Let assumptions~\ref{as:iid} and~\ref{as:links} hold. Define $N_k = f_k N$ and ${\cal G}_k = \{a: F_k(a) \geq 0 \}$.

	(i) For all $b \geq 0$  and all $N$ we have:\eqs{
P^{N}(\pi,b+ K - 1) \leq 1 - \prod_{k=1}^K \left[1 - \min_{a_k \in {\cal G}_k} e^{ N_k F_k(a_k) - a_k b} \right].
}
	(ii) If $R > 1$, we have $a_k^\star > 0$ and for all $N$ and all $b > 0$:
	\eqs{
	  P^{N}(\pi,b + K - 1) \leq 1 - \prod_{k=1}^K [1 - e^{-a_k^\star b}].
	}
	(iii) If $R \leq 1$, and $X_k(\ell)$ is $v_k^2$-sub-Gaussian for all $k$, then for all $b > 0$ and all $N$:
	\eqs{
		P^{N}(\pi, (R^{-1}-1)N + b + K - 1) \leq 1 - \prod_{k=1}^K [1 - e^{ -\frac{b^2}{2 v_k^2 N f_k}}].
	}
\end{theorem}

\begin{corollary}\label{cor:iid_upper}
	The following holds without assumption~\ref{as:links}.

	(i) For all $b \geq 0$  and all $N$ we have:\eqs{
P^{N}(\pi,b+ K - 1) \leq \sum_{k=1}^K \min_{a_k \in {\cal G}_k} e^{ N_k F_k(a_k) - a_k b}.
}
	(ii) If $R > 1$, we have $a_k^\star > 0$ and for all $N$ and all $b > 0$:
	\eqs{
	  P^{N}(\pi,b + K - 1) \leq \sum_{k=1}^K e^{-a_k^\star b}.
	}
	(iii) If $R \leq 1$, and $X_k(\ell)$ is $v_k^2$-sub-Gaussian for all $k$, then for all $b > 0$ and all $N$:
	\eqs{
		P^{N}(\pi, (R^{-1}-1)N + b + K - 1) \leq \sum_{k=1}^K e^{-\frac{b^2}{2 v_k^2 N f_k}}.
	}
\end{corollary}

\noindent{\textbf{Proof.}}\
(i) Consider $n$ such that $\pi_n = k$. Define $M_k(t) = \sum_{\ell=1}^{t} X_k(\ell)$. Consider $d= d_k(n)$, so that $n$ is the $d$-th chunk requested on link $k$. The pre-buffering time is equal to $b + K - 1$, and chunk $n$ is not received in time if and only if $M_k(d) \geq n + b + K - 1$. Further, since $\pi$ is $f$ upper balanced we have $d = d_k(n) \leq (n + K - 1)f_k$ so that $d/f_k + 1 - K \leq n$. It is also noted that $d_k(n) \leq N_k$. Hence, using independence:
\als{
P^{N}(\pi,b +  K - 1) &\le \mathbb{P}\left[\exists k : \max_{1 \leq d \leq N_k} M_k(d) - d/f_k \geq b \right] \\
&\le 1 - \prod_{k=1}^K \left(1 - \mathbb{P}\left[ \max_{1 \leq d \leq N_k} M_k(d) - d/f_k \geq b \right]\right)
}
Consider $a \in {\cal G}_k$. Define $Z_k(d) = e^{a(M_k(d) - d/f_k)}$. $Z_k(d)$ is a sub-martingale, indeed,
\eqs{\mathbb{E}[Z_k(d) | Z_k(d-1) ] = Z_k(d-1) e^{ F_k(a)} \geq Z_k(d-1),}
since $a \in {\cal G}_k$. Using Doob's maximal inequality:
\als{
\mathbb{P}[ \max_{1 \leq d \leq N_k}   M_k(d) - d/f_k \geq b ] &=\mathbb{P}[ \max_{1 \leq d \leq N_k} Z_k(d) \geq e^{a b} ]
													\leq e^{-ab} \mathbb{E}[ Z_k(N_k) ]
													= e^{  N_k F_k(a)  - ab} \\
													&\leq \min_{a \in {\cal G}_k}  e^{ N_k F_k(a) - a b}.
}
Replacing we get the first claim:
\als{
P^{N}(\pi,b + K - 1)
\leq 1 - \prod_{k=1}^K \left[1 - \min_{a_k \in {\cal G}_k} e^{N_k F_k(a_k) - a_k b} \right].
}

(ii) Assume that $R > 1$. For all $k$, we prove that either $X_k(\ell) \leq 1/f_k$ a.s., or otherwise there exists $a_k^\star > 0$ such that $F(a_k^\star) = 0$. The derivative of $F_k(.)$ evaluated at $a = 0$ is $\mathbb{E}[X_k(\ell)] - 1/f_k = \mu_k(1 - R) < 0$. Therefore $F_k(a) < 0$ for $a > 0$ on an open neighborhood of $0$. Choose $\epsilon > 0 $ such that $\mathbb{P}[ X_k(\ell) \geq 1/f_k + \epsilon] > 0$. We have:
\als{
F_k(a) \geq \log( \mathbb{P}[ X_k(\ell) \geq 1/f_k + \epsilon] e^{a (1/f_k + \epsilon)} ) - a /f_k
					\sim a\epsilon \tto{a \to +\infty} +\infty
}
Hence by continuity of $F(.)$ there exists $a_k^\star > 0$ such that $F(a_k^\star) = 0$. The second claim is obtained by setting $a_k \equiv a^\star_k$ for all $k$ in the first claim.

(iii) Consider $R \leq 1$. Define $B = b + N(R^{-1}-1)$. For all $a \geq 0$, from Jensen's inequality we have $G_k(a) \geq a \mu_k$, so $F_k(a) \geq a \mu_k(1 - R) \geq 0$.  Hence $[0,+\infty) \subset {\cal G}_k$ and $a_k \in {\cal G}_k$. Since $X_k(\ell)$ is $v^2_k$-sub-Gaussian, $G_k(a) \leq a \mu_k + a^2 v_k^2/2$, so that:
\als{
N_k F_k(a) - a B \leq a( N(R^{-1}-1) - B) + a^2 v_k^2 N f_k /2
								= - a b + a^2 v_k^2 N f_k /2.
}
Setting $a = {b \over v_k^2 N f_k}$ we obtain $N_k F_k(a) - a B \leq - \frac{b^2}{2 v_k^2 N f_k}$. Substituting in the first claim (where $b$ is replaced by $B$) we obtain the announced result.

The corollary follows from the same reasoning but using the union bound (which does not require assumption \ref{as:links} to hold):
\als{
P^{N}(\pi,b +  K - 1) \le \mathbb{P}\left[\exists k : \max_{1 \leq d \leq N_k} M_k(d) - d/f_k \geq b \right]
\le \sum_{k=1}^K \mathbb{P}\left[\max_{1 \leq d \leq N_k} M_k(d) - d/f_k \geq b \right].
}

\hfill $\Box$

\subsection{Performance for Markovian delays}
We turn to Markovian delays. Even for very simple transition matrices $Q^k$ and rate functions $r(.)$ it is difficult to find a tractable formula for the starvation probability, as it involves upper bounding the crossing probability of a random walk with correlated increments (see for instance \cite{Kugler13}). It is noted that the starvation probability does not only depend on stationary distributions of the link states $m_k$ but also on the the transition rate matrix $Q^k$. One can expect larger starvation probability when the Markov chain has a strong correlation since in that case the streaming flow can experience long bursts of large delays. We propose to consider a regime where $S_k(t)$ evolves on a ``faster time scale'' than the streaming flow of interest. In that case we can find a simple, tractable expression for the starvation probability shown in theorem~\ref{th:markov_upper}. This approximation is useful in practice, since streaming flows are long (several minutes), while most phenomena causing link variability are short, for instance fading, medium access protocols and short lived elastic flows.  We consider speed $\phi > 0$, and we define the accelerated process $( S_k( \phi t) )_{t}$. This process is again a continuous time, stationary ergodic Markov chain, with the same stationary distribution $m_k$, and transition rate matrix $\phi Q^k$.

 Theorem~\ref{th:markov_upper}, statement (i) shows the intuitive fact that when $R > 1$ and one lets $\phi \to \infty$, the starvation probability vanishes. Indeed, seen from the streaming flow, the link variability disappears and the chunk delays become equal (almost surely) to $r_k$ due to the ergodic theorem for Markov chains,  This suggests to look at a setting when $R$ depends on $\phi$ and approaches $1$ as $\phi \to \infty$. To avoid a trivial result (i.e. $P \to 0$ or $P \to 1$ when $\phi \to \infty$), the sum rate and pre-buffering time should scale as $1/(1 - C_1/\sqrt{\phi})$ and $C_2/\sqrt{\phi} + K - 1$ respectively with $C_1,C_2$ two positive constants, as done in theorem~\ref{th:markov_upper}, statement (ii). Before stating our result we state lemma~\ref{lem:battacharya} due to~\cite{Bhattacharya82}, which shows that the amount of data received on a link can be approximated by a Wiener process with appropriate drift and variance. By a slight abuse of notation, we identify $r(.)$ with the vector $(r(i))_{i \in {\cal S}}$.
\begin{lemma}[\cite{Bhattacharya82}]\label{lem:battacharya}
	Define $G^{\phi}(t) = \sqrt{\phi} \int_{0}^t (r( S_k(\phi u))- r_k)du$. Consider $g^k = (g^k(i))_{i \in {\cal S}}$ a solution to the Poisson equation: $Q^k g^k  = r(.) - r_k$. Further define the asymptotic variance :
	\eqs{
	\overline{\sigma}_k^2 =  - 2 \sum_{i \in {\cal S}} r(i) g^k(i) m_k(i).
	}
	Then $G^{\phi}(.)$ converges to a Wiener process with drift $0$ and variance $\overline{\sigma}_k^2$, when $\phi \to \infty$.
\end{lemma}

Given transition matrix $Q^k$, the Poisson equation can be solved numerically by calculating the pseudo inverse of $Q^k$. We show below that for in several cases of interest, one can solve the Poisson equation in closed form and deduce the asymptotic variance $\overline{\sigma}_k^2$. This makes our performance bounds fully tractable. We may now state theorem~\ref{th:markov_upper}. The proof uses standard results on crossing probabilities for Wiener processes, and the interested reader can refer to \cite{Resnick,Billingsley} (for instance).
\begin{theorem}\label{th:markov_upper}
Consider $\pi$ an $f$-upper balanced policy. Let assumptions~\ref{as:markov} and \ref{as:links} hold.

(i) Consider $R > 1$ and $b > 0$ fixed. Then for all $N$:
\eqs{
	 P^{N,\phi}(\pi, b + K - 1) \tto{\phi \to \infty} 0.
}

(ii) Consider $C_1,C_2 \geq 0$, define $b^{\phi} = C_2/\sqrt{\phi}$ and assume that $R = R^{\phi} \equiv 1/(1 - C_1/\sqrt{\phi})$. Then for all $N$:
\als{
	\lim \sup_{\phi \to \infty} P^{N,\phi}(\pi, b^{\phi} + K - 1)
	\leq 1 - \prod_{k=1}^{K} \left( 1 - \mathbb{P}\left[ \max_{t \in [\![ K,N+K-1]\!]} \hspace{-0.5cm} \overline{\sigma}_k W(t) - r_k C_1 t \geq r_k C_2 \right] \right).
}
with $(W(t))_t$ a standard Wiener process.

(iii)(a) If $C_1 = 0$, we have:
\als{
\lim \sup_{\phi \to \infty} P^{N,\phi}(\pi, b^{\phi} + K - 1)
\leq 1 - \prod_{k=1}^{K} \left( 1 -  2 \Psi\left( \frac{r_k C_2}{\overline{\sigma}_k \sqrt{N + K - 1}}\right) \right)
}
(iii)(b) If $C_1 > 0$, for all $N$:
 \eqs{
\lim \sup_{\phi \to \infty} P^{N,\phi}(\pi, b^{\phi} + K - 1) \leq 1 - \prod_{k=1}^{K} \left( 1 - e^{ - 2 \frac{r_k^2 C_1 C_2}{\overline{\sigma}_k^2}} \right).
}
\end{theorem}

\begin{corollary}\label{cor:markov_upper}
 The following holds without assumption~\ref{as:links}.

(i) Consider $R > 1$ and $b > 0$ fixed. Then for all $N$:
\eqs{
	 P^{N,\phi}(\pi, b + K - 1) \tto{\phi \to \infty} 0.
}

(ii) Let $C_1,C_2 \geq 0$, $b^{\phi} = C_2/\sqrt{\phi}$ and $R = R^{\phi} \equiv 1/(1 - C_1/\sqrt{\phi})$. For all $N$:
\als{
	\lim \sup_{\phi \to \infty} P^{N,\phi}(\pi, b^{\phi} + K - 1)
	\leq \sum_{k=1}^{K} \mathbb{P}\left[ \max_{t \in [\![ K,N+K-1]\!]} \overline{\sigma}_k W(t) - r_k C_1 t \geq r_k C_2 \right].
}
with $(W(t))_t$ a standard Wiener process.

(iii)(a) If $C_1 = 0$, we have:
\als{
\lim \sup_{\phi \to \infty} P^{N,\phi}(\pi, b^{\phi} + K - 1) \leq \sum_{k=1}^K 2 \Psi\left( \frac{r_k C_2}{\overline{\sigma}_k \sqrt{N + K - 1}}\right)
}
(iii)(b) If $C_1 > 0$, for all $N$:
 \eqs{
\lim \sup_{\phi \to \infty} P^{N,\phi}(\pi, b^{\phi} + K - 1) \leq \sum_{k=1}^{K}  e^{ - 2 \frac{r_k^2 C_1 C_2}{\overline{\sigma}_k^2}}.
}
\end{corollary}

\noindent{\textbf{Proof.}}\

	(i) We omit superscripts $.^{\phi}$ for clarity. We define $H_k(t) = \int_{0}^t r(S_k(\phi u)) du$, and $G_k(t)= \sqrt{\phi} (H_k(t) - r_k t)$ and $B = b + K - 1$. Consider chunk $n$, to be read at time $n + B$, and requested on link $k$, so that $\pi_n = k$. Chunk $n$ does not arrive in time if and only if less than $d_k(n)$ chunks have been received on link $k$ at time $n + B$, so that $H_k(n + B) \leq d_k(n) \leq f_k( n + K - 1)$ since $\pi$ is $f$-upper balanced. Hence $H_k(n + B) - (n + B) f_k  \leq - f_k b$. Therefore the starvation probability can be upper bounded by:
	\eqs{
	 \mathbb{P}[ \exists k: \min_{n \in [\![ 1,N]\!]}  H_k(n + B) - (n + B) f_k  \leq - f_k b  ].
	}
	Now, from lemma~\ref{lem:battacharya}, we know that, for all $n$,
	\als{
	H_k(n + B) - (n + B)f_k \tto{\phi \to \infty} (n + B) (r_k - f_k)
	= (n + B) r_k(1 - 1/R) > 0,
	} since $R > 1$. We deduce that the starvation probability goes to $0$ when $\phi \to \infty$ as announced.

	(ii) It is noted that $R$, $f_k$ and $B = b^{\phi} + K - 1$ depend on $\phi$, and we omit the superscript $.^{\phi}$ for clarity. Assume that event $\{H_k(n + B) - (n + B) f_k  \leq - f_k b\}$ occurs. Then:
	\eqs{
		G_k(n + B) + \sqrt{\phi}(r_k - f_k)(n + B) \leq - \sqrt{\phi} f_k b
	}
	We have assumed that $f_k = r_k/R = r_k( 1 - C_1/\sqrt{\phi})$ and $b = C_2/\sqrt{\phi}$, so that replacing above yields:
	\eqs{
		G_k(n + B) + r_k C_1 (n + B) \leq - C_2 r_k( 1 - C_1/\sqrt{\phi}).
	}
	So the starvation probability is upper bounded by:
	\als{
	 \overline{P} &\equiv \mathbb{P}[ \exists k: \hspace{-0.3cm} \min_{t \in [\![ K,N + K - 1]\!]} \hspace{-0.3cm} G_k(t) + t r_k C_1  \leq - C_2 r_k( 1 - C_1/\sqrt{\phi})].
	}
	Letting $\phi \to \infty$ and using lemma~\ref{lem:battacharya}:
	\als{
	 	\lim \sup_{\phi \to \infty} \overline{P} &\le \mathbb{P}[ \exists k: \hspace{-0.3cm} \max_{t \in [\![ K,N + K - 1]\!]}  \overline{\sigma}_k W_k(t) - t r_k  C_1  \geq r_k C_2  ] \sk
	 	&= 1 - \prod_{k=1}^K \left( 1-  \mathbb{P}\left[ \max_{t \in [\![ K,N + K - 1]\!]} \hspace{-0.5cm} \overline{\sigma}_k W(t) - t r_k C_1  \geq r_k C_2\right]\right).
	}
	where $W_1(.),...,W_K(.)$ are independent standard Wiener processes, and where we used independence to obtain the last equality.

	(iii)(a) Consider $C_1 = 0$. We recall the reflexion principle.
	\begin{proposition}
	Consider $W(t)$ a standard Brownian motion, then for all $\omega > 0$ and $T \geq 0$:
	$$\mathbb{P}\left( \sup_{0 \leq t \leq T} W(t) \geq \omega\right) = 2 \mathbb{P}(W(T) \geq \omega).$$
	\end{proposition}

	Now using the reflexion principle:
	\als{
	\mathbb{P}  \left[ \max_{t \in [\![ K,N + K - 1]\!]} \overline{\sigma}_k W(t)  \geq r_k C_2\right] &\leq \mathbb{P}  \left[ \max_{t \in [0, N + K - 1]} \overline{\sigma}_k W(t)  \geq r_k C_2\right] \sk
	&= 2 \mathbb{P}\left[  W(N + K - 1)  \geq r_k C_2/ \overline{\sigma}_k\right] \sk
	&= 2 \Psi\left( \frac{r_k C_2}{\overline{\sigma}_k \sqrt{N + K - 1}}\right),
	}
	and replacing in (ii) gives the announced statement.

	(iii)(b) Consider $C_1 > 0$. We first recall a result on the probability that a Wiener process with negative drift hits a positive level.
	\begin{proposition}
	Consider $W(t)$ a standard Brownian motion, $\sigma > 0$, $\mu > 0$ and $\omega > 0$. Then:
	$$\mathbb{P}\left( \sup_{t \geq 0}( \sigma W(t) - \mu t) \geq \omega\right) = e^{- {2 \mu \omega \over \sigma^2}}.$$
	\end{proposition}
	We apply the result above to yield:
	\als{
	\mathbb{P} \left[ \max_{t \in [\![ K,N+K-1]\!]} \hspace{-0.3cm} (\overline{\sigma}_k W(t) - r_k C_1 t) \geq r_k C_2 \right]
	\leq \mathbb{P}\left[ \sup_{ t \geq 0} (\overline{\sigma}_k W(t) - r_k C_1 t)  \geq r_k C_2\right]
	= e^{ - 2 \frac{r_k^2 C_1 C_2}{\overline{\sigma}_k^2}}.
	}
	Replacing in (ii) gives the announced result.

	The corollary is obtained by the same reasoning using a union bound (which does not require assumption~\ref{as:links}):
\als{
	 \mathbb{P}[ \exists k: \min_{n \in [\![ 1,N]\!]}  H_k(n + B) - (n + B) f_k  \leq - f_k b  ]
	 \le  \sum_{k=1}^K \mathbb{P}[ \min_{n \in [\![ 1,N]\!]}  H_k(n + B) - (n + B) f_k  \leq - f_k b].
	}
	\hfill $\Box$

The result is also valid when $R = 1$ (critical case) since $C_1$ can be null. If $C_1 = C_2$ the bound still holds but is not informative.
\section{Case Studies}\label{sec:case_studies}
In this section we specialize our results to several simple models for wired and wireless links. We show how to calculate relevant quantities such as $G_k(.)$ and $\overline{\sigma}^2$. We focus on a particular link and indexes $._{k}$ , $.^{k}$ are omitted for clarity. We consider simple models for tractability. It is noted that as far as CSMA and opportunistic scheduling are concerned, one could consider rate adaptation (at the expense of more complicated expressions) and still derive the cumulant generating function, since for reasonable models, delays follow a phase-type distribution.
\subsection{Wireless links with random access (CSMA-like)}\label{subsec:wifi}
We consider the following model for random access, which is Bianchi's model~\cite{Bianchi00} with one back-off stage. A fixed number of flows compete for access to a link. Each chunk is split into $n_f$ frames. Time is slotted, with $T_s$ the time slot duration, and when the streaming flow attempts to access the link, it is successful with probability $p$. If the attempt is successful, a frame is transmitted during a slot. If the attempt is unsucessful, then the streaming flow waits for a duration uniformly distributed between $0$ and $W T_s$, with $W$ the window size. Given the number of competing flows and the window size, $p$ may be calculated using a fixed point equation as in~\cite{Bianchi00}. The time required to transmit a frame is $Z = T_s(1 + W \sum_{i=1}^G U_i) $ with $G \sim$Geo($p$) and $(U_{i})$ i.i.d. uniform on $[0,1]$ and independent of $G$. Define $a' = a T_s W$. We have $\mathbb{E}[e^{a' U_i}] = \int_0^{1}  e^{a' u} du = h(a')$. $(U_i)_i$ are i.i.d and independent from $G$ so:
\als{
	\mathbb{E}[ e^{a' \sum_{i=1}^G U_i}] &= \sum_{g \ge 0} \mathbb{P}[ G = g ] \left( \prod_{i=1}^g \mathbb{E}[e^{a' U_i}] \right)
	=  \sum_{g \ge 0} p(1-p)^g h(a')^g = {p \over 1 - (1-p)h(a')},
}
if $(1-p) h(a') < 1$ and $=\infty$ otherwise. We deduce:
$$
	\log( \mathbb{E}[ e^{a Z} ]) = T_s a +  \log \left( \frac{p}{1 - (1-p)h(a W T_s)} \right).
$$
The time to transmit a chunk is the sum of $n_f$ i.i.d copies of $Z$, and we deduce the cumulant generating function:
\als{
	G(a) =  n_f \left( T_s a + \log \left( \frac{p}{1 - (1-p)h(a W T_s)} \right) \right) \,\,\, \text{ , } \,\,\,
	 h(a)=  (e^a-1)/a.
}
with $a$ such that $(1-p) h(a W T_s) < 1$.
\subsection{Wireless links using channel-aware scheduling}\label{subsec:cellular}
We consider channel-aware scheduling with a large number of flows. Once again chunks are divided in $n_f$ frames. In each time slot (duration $T_s$), a scheduler chooses the flow whose ratio between instantaneous data rate and expected data rate is maximal and the chosen flow transmits a frame. Define $p$ the inverse of the number of competing flows, the time required to transmit a frame is $Z = T_s(1 + G)$ where $G \sim$ Geo($p$), and the time required to transmit a chunk is a sum of $n_f$ i.i.d copies of $Z$. We deduce the cumulant generating function: \eqs{
G(a) = n_f  \left( a T_s + \log \left( \frac{p}{1 - (1-p)e^{a T_s}} \right) \right).
}
with $a < -\log(1-p)/T_s$. This model is reasonable if the number of flows is large, so that the instantaneous data rate of the chosen user equals the peak rate (the rate achieved with maximal modulation and coding scheme).

\subsection{Wireless ON-OFF channels}\label{subsec:onoff}

 We consider a typical model used in cognitive radio. A link is shared between a primary user and the streaming flow acting as a secondary user. The primary user's activity follows a two-states Markov process, which is independent of the secondary user activity. The streaming flow transmits only when the primary user is not transmitting. The link has two states: state $0$ when the primary user is active and no data is transmitted so that $r(0) = 0$ and state $1$, when the link is available and data is transmitted at rate $r(1)=1$. The transition rate matrix is $Q = \begin{pmatrix} -\beta & \beta \\ \alpha & -\alpha \end{pmatrix}$. The stationary distribution is $m = ( \frac{\alpha}{\alpha + \beta},\frac{\beta}{\alpha + \beta} )$ and the expected data rate is $r = \frac{\beta}{\alpha + \beta}$. The Poisson equation reads $(-r,1-r) = (g^k_1-g^k_0) ( \beta, - \alpha)$ and a solution is: $g^k = ( 0 , \frac{-1}{\alpha + \beta})$. The asymptotic variance is:
\eqs{
\overline{\sigma}^2 = \frac{2 \beta \alpha}{(\alpha + \beta)^3}.
}
As expected, $\overline{\sigma}^2$ is large when $\alpha + \beta$ is small, since in that case the channel state has a strong time correlation.
\subsection{Sharing links with small flows}\label{subsec:small_flows}
The last model we consider is a link shared between the streaming flow and small exponential flows arriving as a Poisson process. Some form of resource sharing is used (for instance fair rate sharing) and when there are $n$ small flows, the streaming flow transmits data at rate $r(n)$. For fair rate sharing we have $r(n) = 1/(1 + n)$. The state of the link $S(t)$ is the number of small flows at time $t$, and follows an M/M/1 process with arrival rate $\lambda$ and service rate $\mu$. Define the load $\rho = \lambda/\mu < 1$. The stationary distribution is $m(n) = \rho^n(1 - \rho)$. The expected data rate is $\overline{r} = \sum_{n \geq 0} r(n) \rho^n( 1 - \rho)$. Define $\overline{R}(n)= r(n) - \overline{r}$. We now solve the Poisson equation.
\begin{proposition}
The asymptotic variance $\overline{\sigma}^2$ is:
\als{
\overline{\sigma}^2 = {2 \over \mu} \sum_{n \geq 0} \sum_{i=0}^{n-1} \overline{R}(n) \overline{R}(i) (\rho^{n} - \rho^{i}).
}
\end{proposition}
\noindent{\textbf{Proof.}}\
By homogeneity, for a fixed value of $\rho$, it sufficient to solve the Poisson equation for $\lambda = 1$, and divide the obtained solution by $\lambda$. For $\lambda = 1$ the Poisson equation reads:
\als{
\overline{R}(0) &=  g(1) - g(0) \sk
\overline{R}(n) &= g(n+1) + \rho^{-1} g(n-1) - (1 + \rho^{-1}) g(n)  \ \ , \ \ n \geq 1.
}
If $g$ is a solution then for all $x \in \mathbb{R}$, $g' = g + x$ is also a solution, which can be checked by inspection. Hence we look for a solution that verifies $g(0) = 0$. We deduce that $g(1) = \overline{R}(0)$. Define $\tilde{g}(n) = g(n) \rho^n( 1 - \rho)$. Multiplying the previous equation by $\rho^{n+1}( 1 - \rho)$ we get:
\eqs{
	\tilde{g}(n+1) - \tilde{g}(n) = \rho^{n+1}( 1 - \rho) \overline{R}(n) + \rho(  \tilde{g}(n) - \tilde{g}(n-1) ).
}
One may readily check that $\tilde{g}(n+1) - \tilde{g}(n) = \rho^n (1-\rho) \sum_{k=0}^n \overline{R}(k) $ satisfies the above recursion, and we deduce $\tilde{g}$ by summing its increments:
\eqs{
\tilde{g}(n) = \tilde{g}(0) + \sum_{i=0}^{n-1} \tilde{g}(i+1) - \tilde{g}(i)
			= \rho \sum_{i=0}^{n-1} \overline{R}(i) ( \rho^{i} - \rho^{n}).
}
Replacing $\tilde{g}$, we obtain the asymptotic variance for $\lambda = 1$:
\als{
\overline{\sigma}^2 = - 2 \sum_{n \geq 0}  \overline{R}(n) \tilde{g}(n) = 2 \rho \sum_{n \geq 0} \sum_{i=0}^{n-1} \overline{R}(n) \overline{R}(i) (\rho^{n} - \rho^{i}).
}
and dividing by $\lambda$ yields the result.\hfill $\Box$
\section{Numerical Experiments}\label{sec:numerical}
We evaluate the numerical performance of the proposed schemes and the accuracy of the various bounds derived above. Throughout this section we consider $N = 3600$ (a video file of 1 hour with chunks of 1 second). We simply simulate the model described in section III, we do not simulate the actual transmission and decoding of chunks. To calculate the starvation probability, we simulate the successive delays on each link $(X_k(\ell))_{k,\ell}$, calculate whether or not starvation occurs and average the result over $10^6$ independent runs.

\subsection{I.i.d. delays}
 We first consider i.i.d delays. On Figure~\ref{fig:iid}(a) we consider two links with i.i.d exponential delays, and we plot the starvation probability as a function of the pre-buffering time, for various values of $R$. Figure~\ref{fig:iid}(b) shows the same for Gaussian delays with variances $0.5$. In both cases our upper bounds are close to the true starvation probabilities, and their accuracy improves when we approach the critical regime $R \approx 1$. The critical regime is the most interesting in practice, since rate adaptation selects a video data rate close to $R$ to ensure maximal quality while avoiding overload.

On Figure~\ref{fig:wifilte} we consider two heterogeneous links. The first link is a CSMA-type link (subsection \ref{subsec:wifi}) with access probability $p_w$ and window size of $4$ time slots. The second link uses opportunistic scheduling (subsection~\ref{subsec:cellular}) with access probability $p_c$. For both links, the slot size is $T_s = 10ms$ and each chunk is made of $n_f = 100$ frames. The video data rate is chosen to ensure that we have $R > 1$ and close to $1$. This scenario represents a streaming flow split between cellular and WiFi links. We present the starvation probability as a function of the pre-buffering times calculated by simulation, our upper bound, and an approximation obtained by replacing the delays by Gaussian delays with the same mean and variance. In the legend, ''S.'' stands for simulation, ''U.'' for Upper Bound and ''G.'' for Gaussian Approximation. The Gaussian approximation is fair, due to the fact that the delay of a chunk is the sum of a large number of independent random variables (central limit theorem).
\begin{figure}
\begin{center}
	\subfigure[Exponential]{\includegraphics[width=0.49\columnwidth]{exponential_iid.eps}}
    \subfigure[Gaussian]{\includegraphics[width=0.49\columnwidth]{gaussian_iid.eps}}
\end{center}
\caption{I.i.d. delays, $R > 1$, starvation probability vs upper bound.} \label{fig:iid}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=1 \columnwidth]{wifi_lte_iid.eps}
\end{center}
\caption{Wifi + cellular links, $R > 1$,  starvation probability vs upper bound.}
\label{fig:wifilte}
\end{figure}
\subsection{Markovian delays}
We turn to Markovian delays. On Figure~\ref{fig:diff}(a) we consider an ON-OFF link (subsection~\ref{subsec:onoff}). We represent the ratio between the asymptotic variance $\overline{\sigma}_k^2$ and the variance of the amount of data received during a unit of time $\int_{0}^{1} r( S_k(u) ) du$, as a function of the transition rate $\alpha$. As expected, when $\alpha$ grows, the Markov chain $S_k(u)$ moves faster, and we approach the asymptotic regime where $\int_{0}^{1} r( S_k(u) ) du$ becomes normally distributed with mean $r_k$ and variance $\overline{\sigma}_k^2$. Further, this happens for reasonably small values of $\alpha$.

Figure~\ref{fig:diff}(b) shows the same for a link shared with small flows using fair rate sharing (subsection~\ref{subsec:small_flows}). The same conclusions hold, so that the variance of $\int_{0}^{1} r( S_k(u) ) du$ approaches $\overline{\sigma}_k^2$ when $\lambda$ grows (the chain moves faster). Furthermore, we see that the convergence is faster for small loads. This is logical since the mixing time of the chain grows with $\rho$: the process $S_k(t)$ has a stronger time correlation for high loads.

On Figure~\ref{fig:starvdiffusion} we plot the starvation probability as a function of the pre-buffering time for two symmetrical ON-OFF links (resp. links with fair rate sharing and load $\rho = 0.7$). We compare the starvation probability to the diffusion approximation suggested in theorem~\ref{th:markov_upper}. In both cases the diffusion approximation is surprisingly accurate, and gives a tractable approximation to an otherwise intractable (to the best of our knowledge) problem.
\begin{figure}
\begin{center}
	\subfigure[ON-OFF links]{\includegraphics[width=0.49\columnwidth]{poisson_on_off.eps}}
    \subfigure[Fair rate sharing]{\includegraphics[width=0.49\columnwidth]{poisson_mm1.eps}}
\end{center}
\caption{Solution to the Poisson equation vs variance of data rates.} \label{fig:diff}
\end{figure}
\begin{figure}
\begin{center}
	\subfigure[ON-OFF links]{\includegraphics[width=0.49\columnwidth]{starvation_on_off.eps}}
    \subfigure[Fair rate sharing, $\rho = 0.7$]{\includegraphics[width=0.49\columnwidth]{starvation_mm1_rho7.eps}}
\end{center}
\caption{Starvation probability vs diffusion approximation.} \label{fig:starvdiffusion}
\end{figure}
\subsection{Real world data}
Finally we present some experiments on real world traces. We store a file of $32$ MB on an HTTP server and, using HTTP byte range requests, we successively request chunks of $100$ kB of this file and measure the corresponding delay. This gives us a trace with $320$ succesive chunk delays. We repeat this $20$ times to obtain $20$ traces. We go through this process twice: first for a wired link (a laptop connected to the Internet through Ethernet, labelled ``Link 1''), and a wireless link (the same laptop connected to the Internet through a WiFi router, labelled ``Link 2''). Links are heterogenous, and we have ${r_1 \over r_2} \approx 2$.

In table~\ref{tab1} we present the absolute value of the autocorrelation of delays
$
{|\mathbb{E}[ (X_k(\ell) - \mu_k) (X_k(\ell + L) - \mu_k)]| \over \sigma_k^2}
$
estimated from our traces for both links, where $L$ denotes the lag. Given link $k$ we estimate this quantity and average the result over traces. Delays are only  weakly correlated, so that assuming i.i.d. delays (Assumption~\ref{as:iid}) seems adequate.

In Figure~\ref{fig8} we present the starvation probability calculated on four arbitrairly chosen traces. Given a trace, for each link we calculate the empirical distribution of delays, then draw $N$ samples with replacement from this distribution and check whether or not starvation has occured. We then estimate the starvation probability by averaging the result over $10^4$ independent trials (curve 'Trace'). Then we compute $(\hat \mu_k,\hat \sigma_k^2)$ the empirical mean and variance of the delays and we compute the starvation probability when delays are i.i.d. Gaussian with mean and variance $(\hat \mu_k,\hat \sigma_k^2)$ (curve 'Gaussian'). Finally we calculate the analytic formula obtained in Theorem~\ref{th:iid_upper}, statement (ii), when delays are i.i.d. Gaussian with mean and variance $(\hat \mu_k,\hat \sigma_k^2)$ (curve 'Analytic'). The three curves are close to each other so that: i.i.d. Gaussian delays provide a simple and robust model, our analytical formulas predict the starvation probability accurately and provide simple, efficient rules to set the prebuffering time. Delays cannot strictly be Gaussian since they are always positive, but the obtained predictions are accurate across all the considered traces.

\begin{table}[!htbp]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
Lag & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7  \\
 \hline
Link 1   &  1.00   &  0.04   &  0.04   &  0.06   &  0.06   &  0.06   &  0.05   &  0.05    \\
 \hline
Link 2   &  1.00   &  0.03   &  0.04   &  0.04   &  0.04   &  0.05   &  0.05   &  0.06   \\
\hline
\end{tabular}
\end{center}
\caption{Real-world data: autocorrelation of delays}\label{tab1}
\end{table}
\begin{figure}
\begin{center}
	\subfigure{\includegraphics[width=0.49\columnwidth]{starvation_trace_1.eps}}
    \subfigure{\includegraphics[width=0.49\columnwidth]{starvation_trace_2.eps}}
	\subfigure{\includegraphics[width=0.49\columnwidth]{starvation_trace_3.eps}}
    \subfigure{\includegraphics[width=0.49\columnwidth]{starvation_trace_4.eps}}
\end{center}
\caption{Real-world data: starvation probability for different traces.} \label{fig8}
\end{figure}
\section{Conclusion}\label{sec:conclusion}
We have investigated streaming over multiple links. We have provided lower bounds on the starvation probability of any policy and proposed simple schemes that approach those limits. For general delay distributions, we have provided tractable upper bounds for the starvation probability of the proposed policies. Our results cover several models of practical interest such as links that employ CSMA or opportunistic scheduling at the packet level, on-off channels and links that use fair rate sharing at the flow level. Numerical experiments demonstrate the accuracy of the proposed bounds and approximations.

\section*{Acknowlegements}
This work has been performed in the framework of the IDEFIX project, funded by the ANR (Agence Nationale de la Recherche) under the contract number ANR-13-INFR-0006.

\bibliographystyle{IEEEtran}

redpajama_set_name: RedPajamaArXiv


text:
Q: Throw checked exception when Enum constant name length is greater than 10 characters? How do I throw an checked exception preventing compiling the code if one of my Enum constant name lenght is greater than 10 characters ?

A: Probably the most practical solution is to use a Java style checker, such as Checkstyle. Then use a build tool, such as Maven, which can prevent a build from succeeding if the style checks fail.
This allows you to then check all manner of things, including enum length. Although you may need to write your own checker for that - it sounds a little specific.

A: You can't enforce such restriction during compile time because the Java Language Specification does not restrict the lengths of the identifiers (source):

A variable's name can be any legal identifier — an unlimited-length sequence of Unicode letters and digits...

(Note: the class file format however does pose a limit which is about 64K but it's not a JLS restriction.)
The most you can do is throw an Exception at runtime if a declared name is too long, for example in the enum's constructor:
enum Fruit {
    APPLE,
    BANANA,
    STRAWBERRIES;

    private Fruit() {
        if (name().length()>10)
            throw new RuntimeException("Too long name: " + name());
    }
}

Also note that Enum constructors can only throw unchecked exceptions.

redpajama_set_name: RedPajamaStackExchange


Hey guys! I hope you've all had a great summer! There's no better way to start of the school year than with a new ODESZA album! I feel like I've been waiting an eternity for this album, and let me tell you, it did not disappoint. This album is ODESZA's third and was released on September 8th. The album showcases the duo's musical abilities by providing haunting vocals and layered beats. The songs combine a chill electronica feel with a variety of elements, including soul which can be heard in the song Across the Room which features artist Leon Bridges. The album almost creates a dream-like, cinematic experience. Many artists are featured on the album including RY X, Regina Spektor, Naomi Wild, and many more. They had previously release the singles Late Night and Higher Ground, which are still a few of my favorites from the album. La Ciudad and Falls are also a few of my new favorites from the album. The duo is currently touring, so be sure to check out if they're coming to a city near you! Also be sure to tune in at WNJR 91.7FM to hear more songs from ODESZA and artists like them!


text:
Hattersheim am Main est une municipalité allemande située dans le land de la Hesse et l'arrondissement de Main-Taunus.

Histoire

Politique et administration

Personnalités liées à la ville
 Anton Flettner (1885-1961), ingénieur né à Eddersheim.
 Hans Weilbächer (1933-2022), footballeur né à Hattersheim am Main.

Jumelages
La ville de Hattersheim am Main est jumelée avec :
  ;
  ;
 .

Références

Ville en Hesse


The inspiration for the original work work is the social inequality in Brazil. The soil is fertile and yet children in the Brazilian countryside are still dying of hunger. And the president who wanted to put an end to that is now in prison.
The limited edition fine art print "Na mesa falta, mas a terra e farta" by Totonho bears the artist's signature, a handwritten edition number and an embossed Certified Art Giclée™ quality mark.
The fine art print "Na mesa falta, mas a terra e farta" by Totonho is made using 310 grams German Etching paper which is a traditional machine-made etching paper. It has an extremely soft surface with a light structure and is very popular amongst graphic designers. This paper is extremely suitable for complex prints and belongs to the absolute top of digital printing materials. German Etching meets the highest standards set by museums and art collectors.
His meticulous work shows both the beauty of the jungle and the devastation of the planet. With his unique colors and extraordinary reflections, he captures the mysterious Brazilian nature in an inimitable way.


"Stop trying to be someone you are not. I have put people in your path to learn from and to help you grow but not for you to become them. I have created you each and everyone different. Do not be afraid or ashamed to be who it is I have made you to be. I have made no mistakes.
My Church is becoming camouflaged. They are so afraid of taking a stand and sticking out like sore thumb they have become more like the world. Their goal has become to be more sociably accepted. They want to blend in, to fit in well with the community. But I have asked them to take a righteous stand, a holy stand against the principalities of darkness in their community.
It is time to put away fear of what others might think or do. It is time to take a stand. Stand for all that you believe in. Raise up standard of righteousness in your home and in your community.
What have you done with all I have given to you? What have you done with all that I have told you to do? Why will you not stand up and be who I have called you to be? Or will you remain blended with everyone else; with no one really knowing who they really are because they are pieces of so many people?
Know what it is that you believe this day. Know what it is that you are to take a stand on. Know who you are in me, who I have made you to be. Do not let fear of not being accepted keep you camouflaged. Your acceptance, your love is in me. I am your strength. I am all you need. I am looking for a people that will rise up and be who I have called you to be. Take a stand. I have given to you all power and strength and authority.
Rise up sons and daughters of God. Rise up and be counted. Rise up, take up the authority I have given each one of you; and each one take back from the enemy the ground he has stolen from you. If you all individually do that he cannot stand. Rise up, put on my breastplate of righteousness."
I kept seeing how hunters wear orange when they go hunting so they can be seen by other hunters, and so they are not mistaken for an animal. That is kind of thing I was seeing here. But I saw him putting on his people a red vest, representing his blood. He is telling us that there is safety in his blood. Each one of us needs to put that vest on individually each day. We cant lean on another's know-how and strength or courage. Every single one of us needs to put on this vest and know our place in him. We much each individually know how much for our lives that blood covers.
Rise up, Take a stand, be all he has made you to be.