text: Constraints on $\Omm$ and $\sigma_8$ in the base-\LCDM\ model from \planck\ temperature and polarization power spectra (red), and the tighter combined constraint with CMB lensing Brighton, Planck 2018 results. VI. X. Combined with temperature, lensing is detected at 40σ4. Gravitational lensing. VIII. VI. Planck 2018 results. Instituto de Astrofísica de Canarias • IAC, The EDGE-CALIFA survey: exploring the role of molecular gas on galaxy star formation quenching, Mapping the stellar age of the Milky Way bulge with the VVV. IX. VIII. Using the Planck cosmic infrared background (CIB) maps as an additional tracer of high-redshift matter, we make a combined Planck-only estimate of the lensing potential over 60% of the sky with considerably more small-scale signal. We find good consistency between lensing constraints and the results from the $\textit{Planck}$ CMB power spectra within the $\rm{\Lambda CDM}$ model. We present measurements of the cosmic microwave background (CMB) lensing potential using the final Planck 2018 temperature and polarization data. Planck 2018 results. ISSN 0004-6361. Constraints on primordial non-Gaussianity: ... Planck 2018 results. Planck 2018 results. Combining with Planck CMB power spectrum data, we measure σ8 to better than 1% precision, finding σ8 = 0.811 ± 0.006. Planck, previamente coñecido como COBRAS-SAMBA, foi un observatorio espacial lanzado o 14 de maio de 2009 mediante un foguete Ariane 5 ECA xunto co observatorio Herschel desde o Centro Espacial de Kourou para estudar a radiación cósmica de fondo de microondas (RCFM). The italian team, after having examined the Planck data to test the gravitational lens effects, not only found that the distortions produced are higher than expected, ... Planck 2018 results. to deliver cooling power to stages at 4.2 and 1.6 K. Stainless steel capillaries facilitate a high ow impedance connection VI. Combined with temperature, lensing is detected at 40σ. • Planck 2018 results. I. Overview and the cosmological legacy of Planck, Astronomy and Astrophysics, 2020; Planck 2018 results. We present measurements of the cosmic microwave background (CMB) lensing potential using the final Planck 2018 temperature and polarization data. High Frequency Instrument data processing and frequency maps, Astronomy and Astrophysics, 2020; Planck 2018 results. Save as MODS; Export to Mendeley; Save as EndNote SCG-2018-08 Planck Collaboration: Planck 2018 results. Roy. CMB from Planck 2018¶. Gravitational lensing • Planck 2018 results. III. High-resolution reddening map, A Systematic Search of Zwicky Transient Facility Data for Ultracompact Binary LISA-detectable Gravitational-wave Sources, https://www.facebook.com/InstitutodeAstrofisicadeCanarias, https://www.instagram.com/iac_astrofisica/, https://www.linkedin.com/company/instituto-de-astrof-sica-de-canarias, https://www.flickr.com/photos/institutodeastrofisicadecanarias/, http://www.idi.mineco.gob.es/portal/site/MICINN, http://www.gobiernodecanarias.org/principal, Planck 2018 results. pp. Using polarization maps filtered to account for the noise anisotropy, we increase the significance of the detection of lensing in the polarization maps from 5σ to 9σ. We increase the significance of the detection of lensing in the polarization maps from $5\,\sigma$ to $9\,\sigma$. Combined with temperature, lensing is detected at 40σ4. The cryostat, one of the largest to have own on a stratospheric payload, uses liquid 4 He. Combined with temperature, lensing is detected at $40\\,σ$. 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# Citation in plain text

I need to cite a paper at the end of a quotation right after the end of the text in beamer. In my example the reference appears after the text, but in italics. How can I change it to the normal font?

• Without a complete minimal working example it's difficult to help. You show us a cake and we must guess the increments. – Marco Daniel Nov 12 '13 at 16:25
• sure sorry ill post it now – datapipe Nov 12 '13 at 17:26

If you use the quote environment with a \cite you can insert \upshape just before the citation, which switches to an upright font (rather than italic).

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# Stratified Pseudomanifold

Hi there, I have a, I guess, simple question. In the definition of an n-dimensional stratified pseudomanifold one demands the following filtration $X=X_n \supset X_{n-1}=X_{n-2} \supset X_{n-3}\supset ... \supset X_0 \supset X_{-1}=\emptyset$. where all $X_i$'s are closed subspaces of X. Why is it required that $X_{n-1}=X_{n-2}$?

Thanks!

Levi

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Levi -- this condition, together with the requirement that the 0-codimensional stratum is connected and orientable, ensures that we can define the fundamental class of $X$. –  algori Jun 3 '11 at 14:05

I think the reason must be that a pseudomanifold $V$ has singular locus $\Sigma V$ of codimension 2 or greater. (Stratifications of varieties are obtained by letting $X_{k-1}$ be the singular locus of $X_k$, or some refinement of that to get the Whitney conditions.) This codimension condition is reflected topologically by the nonbranching condition in the definition of pseudomanifold.

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It's not!

Okay, that's a little too glib. You're right that it's often required that $X^{n-1}=X^{n-2}$, but it depends somewhat on your purpose. In most of my recent papers, including those I'm current working on with Jim McClure, and in the papers of Saralegi (which look at intersection homology from an analytic point of view), there's freedom to include codimension one strata, but you need to be more careful about your definition of intersection (co)homology.

In even more detail: there are a variety of connected reasons why one would classically not work with codimension one strata. One reason has to do with the properties of classical intersection homology theory. Many of the results of intersection homology, including the ones that are approached sheaf theoretically, depend, at heart, on having a nice formula for the intersection homology on cones. If there are no codimension one strata around and you're working with perversities as defined by Goresky and MacPherson, then this cone formula works out, the sheaf of intersection chains is quasi-isomorphic to the Deligne sheaf, and away you go. However, with codimension one strata, and Goresky-MacPherson perversities, the cone formula doesn't quite work out. I have a detailed exposition about this in the following paper:

Greg Friedman, An introduction to intersection homology with general perversity functions, in Topology of Stratified Spaces; Greg Friedman, Eugénie Hunsicker, Anatoly Libgober, Laurentiu Maxim (editors) Mathematical Sciences Research Institute Publications 58, Cambridge University Press 2011, 177-222

Here's the link to the copy on my web site:

http://faculty.tcu.edu/gfriedman/papers/MSRI-revised-2.pdf

One argument that I make there looks carefully at what the perversity should be on such a stratum: if the perversity is 0 (as one might expect), then the perversity is greater than the top perversity allowed by Goresky and MacPherson (which would have to be -1 for a codimension one stratum), and that leads to the cone formula issues. On the other hand, if the perversity assigned is $\leq -1$, now you're below the bottom perversity $\bar 0$ allowed by Goresky and MacPherson, and that causes other issues.

Another way to think about codimension one strata is that they could arise if you took a manifold with boundary and let the boundary be a codimension one stratum. Then, as algori notes, you have a problem getting your fundamental class as a geometrically defined cycle (unless you sufficiently modify your definition of intersection homology). And that causes trouble with Poincare duality.

So, after all that, in summary: a pseudomanifold can have codimension one strata, but much of intersection homology breaks down. Unless you modify your intersection homology a bit, and then everything does work! See my papers or feel free to e-mail if you have further questions.

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# How does one select some $i$ to prove $\exists i$ s.t. $s_i \leq A$ where $A$ is the average of real numbers $(s_1 + s_2 + . . . + s_n) / n$

How does one select some $i$ to prove $\exists i$ s.t. $s_i \leq A$ where $A$ is the average of real numbers $(s_1 + s_2 + . . . + s_n) / n$

It seems like a trivial proof, but writing a proof for such an existence conventionally does not seem obvious to me.

Perhaps I am reading it wrong, but I am sure it says something like: For any real number $A$ which is the average of a set, $S$, of real numbers, there exists a number $s \in S$ s.t. $s$ is less than or equal to $A$.

Is there a special way to choose $i$ to prove this existence? It seems I cannot trivially select a set of numbers and show there is a value that satisfies $s_i \leq A$ since it is some arbitrary set. How would you select $i$ for this proof?

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No, you don't know which $i$ it must be. You go about it as a proof by contradiction. – Calvin Lin Feb 5 '13 at 5:54
Take $s_i$ to be the smallest element in $S$ ($S$ is finite I suppose) – Ewan Delanoy Feb 5 '13 at 5:55

If not, then $s_i> A$ for all $i$. So $$A=\frac{s_1+s_2+\cdots+s_n}{n}>\frac{\overbrace{A+A+\cdots+A}^{n\text{ times}}}{n}=A$$ So derive a contradiction.
So how might I write this then. To yield a contradiction, assume for all $i$, $i > A$. This implies the average of such a series is actually equal to, and not greater than, $A$. And this is the contradiction? Very much as you have written? Do I need to declare $i$ be a natural or something like that? – Leonardo Feb 5 '13 at 6:02
@Leonardo Assume $s_i >A$ for all $i$, then this assumption implies $A>A$ and it implies $A\neq A$. – Hanul Jeon Feb 5 '13 at 6:06


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# Vinoth Nandakumar (University of Sydney)

## Modular representations of $$\mathfrak{sl}_n$$ with two-row nilpotent $$p$$-character

We will study the category of modular representations of the special linear Lie algebra with a central p-character given by a nilpotent whose Jordan type is a two-row partition. Building on work of Cautis and Kamnitzer, we construct a categorification of the affine tangle calculus using these categories; the main technical tool is a geometric localization-type result of Bezrukavnikov, Mirkovic and Rumynin. Using this, we give combinatorial dimension formulae for the irreducible modules, composition multiplicities of the simples in the baby Vermas, and a description of the Ext spaces. This Ext algebra is an "annular" analogues of Khovanov’s arc algebra, and can be used to give an extension of Khovanov homology to links in the annulus. This is joint with Rina Anno and David Yang.


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ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.18 No.2 pp.234-244
DOI : https://doi.org/10.7232/iems.2019.18.2.234

# Strategic Joining Behaviors and Optimal Control for Discrete-Time Passenger-Taxi Matching Queues

Sung Dong Moon, Doo Ho Lee*
Division of Software, Media and Industrial Engineering, Kangwon National University, Gangwon, Republic of Korea
Corresponding Author, E-mail: enjdhlee@kangwon.ac.kr
November 21, 2018 February 25, 2019 March 12, 2019

## ABSTRACT

This study explores strategic joining behaviors and optimal control for discrete-time passenger-taxi matching queues. We analyze strategic joining behaviors by passengers in both observable and unobservable cases. In both cases, we provide the equilibrium joining strategies for individual passengers and optimal joining strategies for socially concerned passengers. We also investigate the optimal taxi buffer size to maximize the social welfare from a governmental perspective. For each social welfare maximization problem, we present a simulated annealing algorithm which helps us find the optimal control scheme for our model. Finally, we conduct various numerical experiments to suggest an insightful operational control methodology.

## 1. INTRODUCTION

It often takes a long time to find a taxi at an airport, ferry terminal or railway station during peak hours. In some very touristy areas, this problem is worse given the increased numbers of passengers and visitors. As Anwar et al. (2013) noted, when too many taxis wait in a taxi queue, it reduces the number of taxis available to service the rest of the city and it also reduces the revenue of taxi drivers waiting in such queues because they forgo the opportunity to earn fares elsewhere. In contrast, when too few taxis are available, passengers are likely to endure long waits. As such, this type of matching issue between passengers and taxis has been an interesting and an important issue in the area of urban transport over the decades. How many taxis do we need at a taxi station so that waiting times are not too long? How much space should the taxi station have? How much should the taxi fare be to maximize social welfare? In this paper, we model this passenger-taxi matching problem as a matching queueing system. Our purpose is to answer the above questions and provide passengers with equilibrium and optimal joining strategies in order to coordinate passengers and taxis while also improving the performance of the system in terms of social welfare.

A matching queue, also known as a double-ended queue, was initially proposed by Kendall (1951), who introduced the matching problem between passengers and taxis. On one side is a queue of taxis and on the other a queue of passengers. A short taxi queue may inconvenience passengers in some cases, but a long taxi queue clearly does while also wasting valuable and expensive space. Moreover, both short and long queues can never coexist. In everyday life, many arrangements in a variety of fields, including communication networks, production inventory systems, and service systems, can be modelled as matching queues. According to Kendall (1951), the size of the queue is caused by the difference between the arrivals of passengers and taxis. The authors also pointed out that the mean queue length approaches zero when the two arrival rates are identical and the variance of the queue length increases infinitely. Dobbie (1961) and Giveen (1963) extended this topic to a matching queue with nonhomogeneous Poisson arrivals of passengers and taxis. Jain (1962) discussed the matching problem under the assumption that the inter-arrival time of taxis is distributed as a general distribution and only a finite capacity of taxis is available. Kashyap (1966) investigated a matching queue while assuming that there is limited waiting space both for customers and taxis, with taxis serving the passengers in bulk. An early discussion with references is given in Srivastava and Kashyap (1982). More recently, the matching queue has been applied to assembly facilities producing multipart components. Parts arrive independently according to a renewal process, but the component can only be assembled when all of the parts are present. Papers that discuss these assembly-like matching queues include those by Som et al. (1994) and Takahashi et al. (2000). Conolly et al. (2002) studied the transient behavior of a double-ended queueing system with state-dependent impatience, also known as reneging or abandonment. The taxi queue is shortened by either passenger arrival or taxi reneging events, and vice versa for a passenger queue. Based on the two-dimensional Markov chain, the authors presented a moment-generating function and first-passage times. Wong et al. (2005) adopted an absorbing Markov chain to model the search process of taxi movements and proposed a useful formulation for describing taxi urban services in a network. Crescenzo et al. (2012) dealt with a matching queue with system failures and repairs, assigning transient and steady-state probability distributions for passenger and taxi queue lengths, respectively. Afèche et al. (2014) studied the matching queue with batch arrivals and abandonment. There are two types of customers, patient ones who queue but may later abandon, and impatient ones who depart immediately if their order is not filled. The authors first established the closed-form results for the double-sided queueing model with batch arrivals and abandonment. More information on the matching queue and actual applications is available in several published studies (Perry and Stadje, 1999;Zenios, 1999;Yang et al., 2010;Yang and Yang, 2011;Wong et al., 2014;Gurvich and Ward, 2014) and the references therein.

Most of the above works focus on determining system performance measures such as queue lengths and waiting times. Although some discussed optimal control schemes, they did not consider strategic joining behaviors between passengers and taxi drivers. Because passengers and taxi drivers are both sensitive to delays, they typically decide whether or not to join a queue based on welldefined joining strategies. This can therefore be modeled as a game among arriving passengers and taxis, and the fundamental problem is to identify the Nash equilibrium. Naor (1969) was the first to show that tolls can be levied to encourage strategic customers to adopt socially optimal behavior that may not be optimal for them personally. Since then, a number of studies of strategic joining/balking behavior by customers in queueing systems have been conducted. The monograph of Hassin and Haviv (2003) summarize the main approaches and several results on various observable and unobservable queueing systems with extensive bibliographical references. Recently, equilibrium joining/balking strategies in matching queues have been the subject of growing attention. Li et al. (2016) considered a make-to-stock production system with setup costs and delay-sensitive customers. They modeled the production system as a matching M/M/1 make-to-stock queue with a two-critical-number policy. Under this policy, production starts when N customer orders have accumulated and stops immediately when the inventory level reaches S. Li et al. (2016) derived the equilibrium customer purchasing strategies and system performance metrics. Those authors also obtained the optimal values of two control variables. Shi and Lian (2016a) analyzed the strategic joining behavior of passengers in both the observable and the unobservable cases. In the observable case, those authors provided the individual equilibrium and the socially optimal joining strategies of threshold type. On the other hand, in the unobservable case, they discussed those of probability type. Further, the optimal taxi buffer size which maximizes the overall social welfare was determined from the perspective of a government (or a social planner). Shi and Lian (2016b) extended their work to the equilibrium strategies and optimal control method in a matching queue where there is limited waiting space both for passengers and taxis. They considered a centralized system and a decentralized system. In the centralized system, the government sets thresholds for both passengers and taxis to maximize social welfare. On the other hand, in the decentralized system, passengers and taxis determine whether to join a queue or not based on their individual utility functions. More recently, Wang et al. (2017) analyzed the strategic behavior and social optimization in a double-ended queue with gated policy. In Wang et al. (2017)’s model, taxis are not allowed to enter if the number of taxis reaches an upper limit (no waiting passengers) and are allowed to enter again when the number of waiting taxis is reduced to a lower limit.

Motivated by the above excellent works, this paper explores equilibrium joining strategies and social welfare maximization in passenger-taxi matching queues under a discrete-time setting. For decades, the discrete-time queue has received much attention due to its actual use in timeslotted digital communication systems and other synchronous systems (Lee, 2016), as the discrete-time queue is more suitable than the continuous-time queue for modeling systems in which the basic operational units are bits, packets and cells. The practical interest in discrete-time matching queues is not confined to taxis and passengers. The lack of radio spectrum resources has limited the sustainable development of wireless communication technologies and mobile application networks. The cognitive radio network (CRN) is the most promising candidate solution to address such spectrum shortages and inefficiencies. The main principle of a CRN is that the secondary user is intelligently aware of the environment and opportunistically occupies the ‘spectral hole’ of the privileged user. Comparing a CRN with a matching queue, the packet and the channel can be regarded as the passenger and the taxi, respectively. In a CRN, the buffer capacity of arriving packets and the number of available channels should be optimized so that spectrum allocation is realized dynamically and the overall operational profit (the energy consumption) is maximized (minimized). Due to its synchronized features, many studies have assumed discrete-time geometric distributions for packet arrival, packet transmission, packet dropping, packet retrial, channel occupation and other packet processes. By doing so, the discrete-time queue can describe coincidence events and provide the more precise performance measures of various communication systems, including CRNs (Jin et al., 2013;Jin et al., 2016;Jin and Sun, 2017;Zhao and Bai, 2017). To the best of our knowledge, no works thus far have analyzed strategic behaviors in a discrete-time matching queue. In this paper, we extend the work of Shi and Lian (2016a) to a discrete-time matching queue and depict the equilibrium joining/balking strategies of passengers as well as the optimal strategies which maximize social welfare.

The remainder of the paper proceeds as follows. In section 2, we describe the mathematical model and associated assumptions. In section 3, we derive the stationary distribution and the important performance measures. In section 4, we discuss equilibrium joining strategies used by individual and socially concerned passengers in observable and unobservable cases. We also determine the optimal buffer size from a governmental perspective. Section 5 describes the various numerical experiments conducted here and provides an insightful operational control strategy.

## 2. MATHEMATICAL MODEL AND ASSUMPTIONS

The queueing system examined under this study has the following features. Passengers arrive in groups according to a Bernoulli process with a probability of λ (0 < λ < 1). Meanwhile, taxis arrive sequentially according to a Bernoulli process with a probability of μ (0 < μ < 1). For the system to be stable, the inequality λ < μ should hold. In a discrete-time queueing system, arrivals by a taxi and a passenger can occur at the same time at the boundary of the time axis, making it necessary to determine the order of their arrivals. If a taxi and a passenger arrive at the taxi station simultaneously, it is assumed that the taxi arrives instantaneously first. Each group of passengers consists of one to five persons who can be precisely taken to their destinations by a taxi. For convenience, we regard groups of passengers as a single passenger. The passengers and taxis are queued on a firstin- first-out basis. The waiting space for taxis is finite. We define K (K ≥ 1) as the maximum number of taxis that can be accommodated at the taxi station. Because the space occupied by passengers is much smaller than a taxi, the waiting space for passengers is assumed to be infinite.

We assume that a passenger pays a taxi fare of p1 (p1 > 0) on average and receives a reward of R (R > 0) per unit time for taking a taxi. Moreover, taxis and passengers accrue waiting costs while they remain at the taxi station. Here, we define C1 (C1 > 0) and C2 (C2 > 0) as the passenger’s waiting cost per unit of time and the taxi’s waiting cost per unit of time, respectively. Each taxi spends an average of Cf (Cf > 0) on maintenance and fuel costs. Finally, to improve the quality of the taxi service and to reduce traffic congestion, the government can adjust the taxi arrival rate by imposing a tax or granting a subsidy to each taxi. We define p2 as the tax or subsidy affecting the taxi during each trip. p2 > 0 indicates a government grants subsidy to each taxi, while p2 < 0 denotes a government levy on each taxi.

## 3. QUEUEING ANALYSIS

Define $N ( t )$ as the number of the passengers or the taxis at the taxi station at time t, where N(t) > 0 indicates that the passengers are waiting for the taxis, N(t) < 0 denotes that the taxis are waiting for the passengers, and N(t) = 0 means that the taxi station is empty. Because the maximum number of taxis in the system is equal to K and the waiting space for passengers is assumed to be infinite, a stochastic process {N(t), t = 0, 1, 2, ⋯} becomes a one-dimensional discrete-time Markov chain with the state space expressed as Ω ={−K, −K + 1, ⋯, −1, 0, 1, ⋯} and its non-zero one-step transition probabilities are given by

$q − K , − K + 1 = λ , q n , n + 1 = λ μ ¯ , n ≥ − K + 1 , q n + 1 , n = λ ¯ μ , n ≥ − K , q − K , − K = λ ¯ , q n , n = λ μ + λ ¯ μ ¯ , n ≥ − K + 1$

where $λ ¯ = 1 − λ and μ ¯ = 1 − μ$. The corresponding transition diagram is shown in Figure 1.

Let πn be the stationary probability of N(t), which means that $π n = lim t → ∞ Pr { N ( t ) = n }$, with $n ≥ − K$. Then, the local balance equations governing the system are then given by

$π − K λ = π − K + 1 λ ¯ μ$
(1)

$π n λ μ ¯ = π n + 1 λ ¯ μ , n ≥ − K + 1$
(2)

Solving these equations recursively, we have

$π − K = 1 − ρ$
(3)

$π n = ( 1 − ρ ) ω n + K μ ¯ , n ≥ − K + 1$
(4)

where $ω = ρ μ ¯ / λ ¯ and ρ = λ / μ$. From (3) and (4), we calculate the mean passenger queue length L1 and the mean passenger waiting time W1:

$L 1 = ∑ n = 1 ∞ n π n = λ λ ¯ ω K μ − λ$
(5)

$W 1 = L 1 λ = λ ¯ ω K μ − λ$
(6)

From the BASTA property (Takagi, 1993), the blocking probability of an arriving taxi is $π − K$; thus, the effective arrival probability of a taxi μeff is given by $μ e f f = μ ( 1 − π − K ) = λ$. Therefore, we calculate the mean taxi queue length L2 and the mean taxi waiting time W2:

$L 2 = ∑ n = − K − 1 ( − n ) π n = K − λ λ ¯ ( 1 − ω K ) μ − λ$
(7)

$W 2 = L 2 μ e f f = K λ − λ ¯ ( 1 − ω K ) μ − λ$
(8)

## 4. OPTIMIZATION

In this section, we discuss passenger’s equilibrium joining strategies and find the optimal taxi buffer size to maximize overall social welfare. From (6) and (8), the (expected) utility functions of a passenger and a taxi are

$U 1 = R − p 1 − C 1 W 1$
(9)

$U 2 = p 1 + p 2 − C f − C 2 W 2$
(10)

### 4.1 Passenger Strategies

Upon arrival, every passenger decides whether to join the queue or balk. In the rest of paper, it is assumed that $R ≥ p 1 + C 1 / μ$, which ensures that the reward exceeds the total cost for an arriving passenger who finds the empty system. We denote Λ as the arrival rate of passengers

#### 4.1.1 Observable Case4.1.1 Observable Case

In the observable case, an arriving passenger is in-formed about the passenger queue length; hence there exist the equilibrium joining strategy of threshold type. A pure threshold strategy is specified by ne, and the join-ing/balking strategy has the form of ‘While arriving, observe the passenger queue length; join if the passenger queue length is smaller than or equal to ne and balk otherwise.’ Based on the reward-cost structure, for an arriving passenger who decides to join the queue, his/her utility is expressed as $U 1 o b ( n ) = R − p 1 − C 1 T ( n )$, where T(n) is the mean waiting time of the passenger who joins the queue, given that he/she encounters n passengers who are already existing. For T(n), it is clearly equal to the n+1 inter-arrival times of taxis; that is, $T ( n ) = ( n + 1 ) / μ$. The passenger’s utility after taking a taxi in equilibrium then equals to 0; i.e., $U 1 o b ( n ) = R − p 1 − C 1 ( n + 1 ) / μ = 0$. Solving $U 1 o b ( n ) = 0$ with respect to n, we obtain

$n e = μ ( R − p 1 ) C 1 − 1$
(11)

where the symbol $⌊ ⌋$ is a floor function.

Next, we consider that the passengers are socially concerned individuals. Socially concerned passengers who care about society will set an optimal buffer (the socially optimal threshold) for themselves to maximize social welfare. In order to maximize social welfare, we derive a stationary queue length distribution, where we define ns as the social joining strategy to be used. Therefore, the state space of the discrete-time Markov chain is . The related transition diagram is depicted in Figure 2

Define the stationary probability as $φ n = lim t → ∞ Pr { N ( t ) = n }$ for $n ∈ Ω s$. Then, the local balance equa-tions governing the system are given by

$φ − K Λ = φ − K + 1 Λ ¯ μ$
(12)

$φ n Λ μ ¯ = φ n + 1 Λ ¯ μ , − K + 1 ≤ n ≤ n s − 1$
(13)

where $Λ ¯ = 1 − Λ$. Solving (12) and (13) recursively using the conventional Birth-Death process (Ross, 1996), we have

$φ − K = 1 − σ 1 − σ γ n s + K$
(14)

$φ n = γ n + K ( 1 − σ ) μ ¯ ( 1 − σ γ n s + K ) , − K + 1 ≤ n ≤ n s$
(15)

where $σ = Λ / μ$ and $γ = σ μ ¯ / Λ ¯$ . Let bp be the blocking probability of an arriving passenger. According to our assumption with regard to the arrival order of passengers and taxis in section 2, we have $b p = μ ¯ φ n s$; thus, the effective arrival probability of a passenger α is given by

$α = Λ ( 1 − b p ) = Λ 1 − γ n s + K ( 1 − σ ) 1 − σ γ n s + K = Λ 1 − γ n s + K 1 − σ γ n s + K$
(16)

Let bt be the blocking probability of an arriving taxi. According to the BASTA property, we have $b t = φ − K$; thus, the effective arrival probability of a taxi β is given by

$β = μ ( 1 − b t ) = μ 1 − 1 − σ 1 − σ γ n s + K = Λ 1 − γ n s + K 1 − σ γ n s + K$
(17)

With (14) and (15), the mean queue lengths for a passenger and a taxi are respectively expressed as

$L 1 o b = ∑ n = 1 n s n φ n = γ K + 1 ( 1 − σ ) μ ¯ ( 1 − γ ) 2 1 − γ n s ( 1 + n s − n s γ ) 1 − σ γ n s + K$
(18)

$L 2 o b = ∑ n = − K − 1 ( − n ) φ n = 1 − σ 1 − σ γ n s + K K − γ ( 1 − K + K γ − γ K ) μ ¯ ( 1 − γ ) 2$
(19)

It follows that the social welfare function is equal to

(20)

Define $n s *$ as the socially optimal threshold strategy which maximizes the social welfare:

$n s * = arg max n s ∈ ℤ + S W o b ( n s )$
(21)

where $ℤ +$ is the positive integer set. Hence, the optimization problem in (21) becomes a nonlinear integer pro-gramming (NLIP) problem. The maximum solution $n s *$ should satisfy the following inequalities:

$S W o b ( n s * ) − S W o b ( n s * + 1 ) ≥ 0 a n d S W o b ( n s * ) − S W o b ( n s * − 1 ) ≥ 0$
(22)

or equivalently,

$n s * ( 1 − γ ) − σ γ K ( 1 − γ n s * ) ≤ M C 1 γ ≤ ( n s * + 1 ) ( 1 − γ ) − σ γ K ( 1 − γ n s * + 1 )$
(23)

where $M = Λ μ ¯ ( 1 − γ ) 2 ( R + p 2 − C f ) + C 2 γ [ K μ ¯ ( 1 − γ ) 2 − γ ( 1 − K + K γ − γ K ) ]$. From (23), it is clear that, when $Λ = μ or γ = 1$, there is no socially optimal threshold. Define δ(n) as a function of n such that $δ ( n ) = n ( 1 − γ ) − σ γ K ( 1 − γ n ) ∀ n ∈ { 0 } ∪ ℤ +$. Because

$δ ( n + 1 ) − δ ( n ) = ( 1 − γ ) + σ γ K + n ( 1 + γ ) > 0$
(24)

the function δ(n) is the increasing in n. If $δ ( 0 ) ≤ M / C 1 γ ≤ δ ( 1 ) , n s = 0$ but it is not optimal because $n s ∈ ℤ +$. Therefore, if $M / C 1 γ ≥ δ ( 1 )$, there exist a unique $n s *$ which satisfies inequalities in (22) or (23). In addition, it is easy to show that, if $M / C 1 γ ≤ δ ( n e ) ,$ then $n s * ≤ n e$; otherwise, $n s * > n e$. Finally, as $n s → ∞$, we have

$lim n s → ∞ S W o b ( n s ) = Λ ( R + p 2 − C f ) − C 1 γ K + 1 ( 1 − σ ) μ ¯ ( 1 − γ ) 2 − C 2 ( 1 − σ ) K − γ ( 1 − K + K γ − γ K ) μ ¯ ( 1 − γ ) 2$
(25)

Thus, we can set the upper bound of $n s *$ as follows:

(26)

where ε is a sufficiently small real number.

To solve (21), we introduce a simulated annealing (SA) algorithm. SA is a probabilistic search method that was introduced by Metropolis et al. (1953). Later, this method was successfully applied to solve optimization problems by Kirkpatrick et al. (1983). For more infor-mation about SA, readers may refer to Aarts and Korst (1997) and Cercignani (1998).

In Pseudocode 1, we present our SA algorithm for finding $n s *$ which maximizes the social welfare. From lines 1 to 5, we set the input parameters and generate the initial state within the search space . The explanation of the input parameters and the states are given below.

• T : temperature

• κ : cooling coefficient

• Itr : maximum number of iteration

• sbest : best stat

• scurrent : current state

• snew : neighbor state

From lines 6 to 19, the SA algorithm execution pro-cedure is described. In line 7, the cooling schedule is shown to be $1 − κ t$, where t is an iterative index. In practice, the cooling coefficient κ usually falls within 0 < κ < 0.3. In this paper, κ is set to 0.01. After the ran-dom neighbor state is generated and its social welfare value is calculated in line 8, we update the current state and its social welfare value from line 9 to line 12 based on the Metropolis criterion. We also check if the current state is the best from lines 13 to 16. Finishing the loop lines 6 to 17, we finally obtain $n s *$ and $S W o b ( n s * )$. When we assume that $R = 100 , p 1 = p 2 = 10 , C 1 = C 2 = 5 , C f = 30 ,$ $Λ = 0.5 , μ = 0.55 , and K = 10$, we obtain the socially optimal threshold strategy $n s * = 6$ with the individual equilibrium threshold strategy ne = 8. We observe that $n s * ≠ n e$, which indicates that the individual threshold strategy does not always match the socially optimal threshold strategy.

#### 4.1.2 Unobservable Case

In the unobservable case, an arriving passenger does not have any information about the passenger queue length when they decide to join or balk. The optimal decision of an arriving passenger must consider the strategies of the other passengers. Because all passengers are assumed to be indistinguishable, we consider the situation as a symmetric game among them. In the present model, there are only two pure strategies: to join or to balk. The strategy is specified by the joining probability of an arriving passenger. In the unobservable case, our goal is to identify the Nash equilibrium joining strategy of the probability type.

Suppose that all passengers follow a joining strategy Λ1. Then, the system behaves as the original, but with arrival probability is Λ1 instead of λ. For the system to be stable, we assume that $Λ 1 < μ$. The corresponding transition diagram is given in Figure 3.

We consider individual passengers. From (6), the mean passenger waiting time is expressed as

$W 1 ( Λ 1 ) = Λ ¯ 1 ω 1 K μ − Λ 1$
(27)

where $ω 1 = ρ 1 μ ¯ / Λ ¯ 1 and ρ 1 = Λ 1 / μ .$ Based on the reward-cost structure, for an arriving passenger who decides to join the queue, his/her utility is

$U 1 u n ( Λ 1 ) = R − p 1 − C 1 W 1 ( Λ 1 )$
(28)

According to (28), $U 1 u n ( Λ 1 )$ depends on Λ1. We can determine the Nash equilibrium strategy of an arriving passenger Λe. To do this, we establish the three cases below.

• Case 1:$R ≤ p 1 + C 1 W 1 ( 0 )$. In this case, even if no other passengers join, the utility of a passenger who joins the queue is non-positive. Therefore, the joining strategy with probability Λe = 0 is an equilibrium strategy, and no other equilibrium is possible. Moreover, in this case, not joining is the dominant strategy.

• Case 2:$R ≥ p 1 + C 1 W 1 ( Λ 1 )$. In this case, even if all potential passengers join, they all enjoy a non-negative benefit. Therefore, the strategy of joining with probability $Λ e = Λ 1$ is an equilibrium strategy, and no other equilibrium is possible. Moreover, in this case, joining is the dominant strategy.

• Case 3:$p 1 + C 1 W 1 ( 0 ) < R < p 1 + C 1 W 1 ( Λ 1 )$. In this case, if $Λ e = Λ 1$, then a passenger who joins suffers a negative benefit. Hence, this cannot be an equilibrium strategy. Likewise, if Λe = 0, a customer who joins obtains a positive benefit, more than by balking. Hence, this cannot be an equilibrium strategy. Therefore, there exists unique equilibrium joining probability Λe satisfying for which customers are indifferent with regard to whether they join or balk.

Next, we consider the case in which passengers are socially concerned. From (9) and (10), the social welfare function can be given by

(29)

Where $μ e f f = μ ( 1 − π − K ) = Λ 1$ according to the BASTA property. Define $Λ 1 *$ as the socially optimal joining strategy which maximizes the social welfare:

$Λ 1 * = arg max Λ 1 ∈ ( 0 , μ ) S W u n ( Λ 1 )$
(30)

Therefore, equation (30) becomes a nonlinear pro-gramming problem. The objective function in (30) is differentiable with respect to Λ1. Its second-order derivative is

$∂ 2 S W u n Λ 1 ∂ Λ 1 2 = − ω 1 K C 1 + C 2 K 2 μ − Λ 1 2 + K μ − Λ 1 Λ 1 μ ¯ + Λ ¯ 1 Λ Λ 1 Λ ¯ 1 μ − Λ 1 3 − 2 μ μ ¯ μ − Λ 1 3 ω 1 K C 1 + C 2 − C 2$
(31)

Investigating equation (31), it is possible that $∂ 2 S W u n ( Λ 1 ) / ∂ Λ 1 2 < 0 or ∂ 2 S W u n ( Λ 1 ) / ∂ Λ 1 2 > 0$ or depending on the value of $ω 1 K ( C 1 + C 2 ) − C 2 .$ For example, when $R = 1000 , p 2 = 10 , C 1 = 1 , C 2 = 50 , C f = 10 , μ = 0.5 ,$ and $K = 10 , ∂ 2 S W u n ( Λ 1 ) / ∂ Λ 1 2 Λ 1 = 0.4 = 19030 > 0$ and $∂ 2 S W u n ( Λ 1 ) / ∂ Λ 1 2 Λ 1 = 0.49 = − 299812 < 0$. Therefore, we cannot determine the convexity/concavity of $S W u n ( Λ 1 )$. To solve (30), we also use the SA algorithm. The SA algorithm to maximize $S W u n ( Λ 1 )$ is described in Pseudocode 2; it is fairly similar to Pseudocode 1 apart from several lines. Hence, we omit the details of Pseudocode 2 here. When we assume that $R = 100 , p 1 = p 2 = 10 , C 1 = C 2 = 5 , C f = 30 , μ = 0.55 ,$ and $K = 10 ,$ we obtain the socially optimal joining strategy of $Λ 1 * = 0.5118 ,$ with the individual equilibrium joining strategy of $Λ e = 0.5356.$ We observe that $Λ 1 * ≠ Λ e$, which indicates that the individual joining strategy is not always consistent with the socially optimal joining strategy.

### 4.2 Government Strategies

We will study the problem of finding an optimal taxi buffer size to maximize the social welfare and guarantee non-negative passenger/taxi utility. Following (9) and (10), the social welfare function is as follows:

(32)

At this point, we can establish the following NLIP problem to maximize the social welfare with respect to K:

(33)

In equation (33), we want to maximize the govern-ment’s social welfare. The first and second constraints guarantee the non-negative passenger and taxi utility outcomes, respectively.

Next, we address the feasibility of (33). It is easy to show that U1(K) means an increase in because $0 < ω < ρ < 1$ and

$U 1 ( K + 1 ) − U 1 ( K ) = C 1 λ ¯ ω K ( 1 − ω ) μ − λ > 0 , ∀ K ∈ ℤ +$
(34)

Thus, if we assume that $U 1 ( 1 ) ≥ 0 ,$ the first con-straint in (33) is always satisfied. Simplifying $U 2 ( K + 1 ) − U 2 ( K )$, we have

$U 2 ( K + 1 ) − U 2 ( K ) = C 2 ω K μ − 1 λ < C 2 ω μ − 1 λ = − ( 1 − ρ ) ( 1 + ρ μ ¯ ) λ λ ¯ < 0 , ∀ K ∈ ℤ +$
(35)

Therefore, U1(K) is a decreasing function of K and there exists an upper bound Kupper which satisfies the following inequality:

(36)

Assuming that $U 2 ( 1 ) ≥ 0$, the second constraint in (33) is always satisfied within the interval of . Summarizing the above analysis, the inequalities should hold for the NLIP problem in (33) to be feasible, and we can rewrite (33) as

$K * = arg max K ∈ S g S W g ( K )$
(37)

where K* is the socially optimal buffer size which maximizes the social welfare and Sg is the search space . To solve (37), we suggest the SA algorithm in Pseudocode 3. Because it is similar to Pseudocodes 1 and 2, we also omit the details of Pseudocode 3 here for brevity. Setting $R = 150 , p 1 = 30 , p 2 = 10 , C 1 = C 2 = 5 , C f = 5 , C f = 10 , λ = 0.6 ,$ and $μ = 0.62$, we obtain the socially optimal buffer size K* = 8. Additionally, the passenger and taxi utility values are $U 1 ( 8 ) = 68.97 and U 2 ( 8 ) = 12.30$, which are both non-negative numbers.

## 5. NUMERICAL EXAMPLES

In this section, we present several numerical exam-ples with which to explore the equilibrium and the opti-mal behavior of passengers in the observable and the unobservable cases and the optimal taxi buffer size from a governmental perspective.

• Example 1. This example deals with the equilibrium joining strategy of an individual passenger and the optimal joining strategy of a socially concerned passenger in the observable case. Initially, we conduct a sensitivity analysis of the taxi arrival probability μ while assuming that $R = 100 , p 1 = p 2 = 10 , C 1 = C 2 = 5 , C f = 30 , Λ = 0.5 ,$ and $K = 10$. Varying the value of μ from 0.4 to 0.6, we determine the equilibrium joining strategy ne and the optimal joining strategy $n s *$, as shown in Figure 4. We observe that both ne and $n s *$ increase monotonically as μ increases. Therefore, as taxis arrive more frequently, it is desirable to raise the passenger joining threshold in terms of both the individual utility of passengers and the overall welfare of society. We also find that $n s * ≤ n e$ because the inequality $M / C 1 γ < δ ( n e )$ holds within the interval .

Next, we consider the passenger arrival probability Λ. Here, we assume that R = 100, p1 = p2 = 10, C1 = C2 = 5, Cf = 30, μ = 0.5, and K = 10. Varying the value of from 0.4 to 0.6, we determine $n e$ and $n s *$ , as pre-sented in Figure 5. According to equation (11), is not a function of ; thus, $n e$ is always equal to $⌊ 0.5 × ( 100 − 10 ) / 5 ⌋ − 1 = 8$ . On the other hand, $n s *$ decreases monotonically as increases, which implies that as passengers arrive more frequently, the passenger joining threshold should decrease to maximize social welfare overall.

• Example 2. In this example, we investigate the equilibrium joining strategy of individual passengers and the optimal joining strategy of socially concerned passengers in the unobservable case. Initially, we conduct a sensitivity analysis of the taxi arrival probability $μ$ , where we assume that $R = 300 ,$ $p 1 = p 2 = 10 ,$ $C 1 = C 2 = 5 ,$ $C f = 30 ,$ and $K = 5.$ Varying the value of from 0.1 to 0.9, we determine the equilibrium joining strategy $Λ e$ and the optimal joining strategy $Λ 1 *$ , as indicated in Figure 6. We find that both Λe and $Λ 1 *$ increase as μ increases. Therefore, it is advantageous for passengers to arrive at the taxi station frequently in order to increase the overall social welfare as well as the utility of individual passengers when a taxi arrives frequently at the taxi station. We also find that $Λ 1 * < Λ e$, which means that the socially optimal joining probability is lower than the individual equilibrium joining probability. Finally, we note that $Λ 1 * < μ and Λ e < μ$ because the system should remain stable.

Next, we examine the effect of the taxi buffer size K on the passenger joining probabilities. We set $R = 300 , p 1 = p 2 = 10 , C 1 = C 2 = 5 , C f = 30 , and μ = 0.5$ Varying the value of Λ from 0.4 to 0.6, we define the values of Λe and $Λ 1 *$, as indicated in Figure 7. This outcome shows that both Λe and $Λ 1 *$ increase as K increases. If the taxi station can accommodate more taxis, more passengers will join the queue to use the taxi service, which benefits not only passengers but also social welfare overall. Figure 7 also confirms that $Λ 1 * < Λ e$, but as the taxi buffer size increases, $Λ 1 *$ grad-ually approaches Λe.

• Example 3. In this example, we analyze the relationship between the optimal taxi buffer size the passenger/ taxi arrival probabilities. Here, we assume that $R = 100 , p 1 = 30 , p 2 = 10 , C 1 = C 2 = 5 , C f = 10 , and λ = 0.48$. Varying the value of μ from 0.5 to 0.8, we obtain the optimal taxi buffer size K* and the maximum social welfare passenger $S W g ( K * )$ , as presented in Figure 8. When μ increases, K* decreases but $S W g ( K * )$ increases. This phenomenon can be explained as follows. As taxis arrive frequently, there are many taxis available at the taxi station; hence, the government does not need to reserve a large number of waiting spaces for taxis. This causes the overall social welfare to increase because the mean taxi waiting time decreases. In addition, when taxis arrive frequently, the mean passenger waiting time is reduced, which has a positive effect on social welfare.

Next, we consider the impact of the passenger arrival on the taxi buffer size, where we set R = 100, p1 = 30, p2 = 10, C1 = C2 = 5, Cf = 10, and μ = 0.81. Varying the value of λ from 0.65 to 0.8, we determine the values of K* and $S W g ( K * ) ,$ as shown in Figure 9. As expected, K* increases when λ increases because, as passengers arrive more frequently, more of a taxi buffer must be provided at the taxi station to reduce passenger waiting times. It is interesting to note that $S W g ( K * )$ initially increase and then sharply decreases after reaching its peak value. When λ = 0.7375, $S W g ( K * )$ reaches its highest value of 65.5042. This can be explained as follows. As passengers arrive frequently, the taxi buffer size should increase, which leads to an increase in the mean taxi waiting time. Therefore, for a large taxi buffer size, social welfare decreases.

## 6. CONCLUSION

In this work, we studied equilibrium joining strate-gies and the social welfare maximization problem in the discrete-time passenger-taxi matching queues. We ana-lyzed passenger strategic joining behaviors in observable and unobservable cases. In the observable case, we provided an equilibrium threshold-type joining strategy for individual passengers and the optimal joining strategy for socially concerned passengers. On the other hand, in the unobservable case, the probability type of strategy was determined and presented. From a governmental perspective, we established the optimal taxi buffer size which maximizes the overall social welfare while also guaranteeing non-negative passenger/taxi utility values. For all social welfare maximization problems, as the taxi fare p1 represents the transfer between passengers and taxi drivers, the objective function does not contain the terms of p1. We also conducted various numerical experiments to offer an insightful operational control strategy.

This paper analyzes passenger equilibrium and op-timality issues. There are various avenues for future research. One can conceive of the strategic and optimal joining behaviors of taxis in both observable and unob-servable cases. Given that taxi drivers are also an im-portant aspect of this system, studying this topic would be a worthy endeavor. Another interesting extension would be to extend our model to the system failure-repair mechanism and to state-dependent passenger/taxi arrivals.

An unrealistic assumption made in this study is the instantaneous service time. In reality, the loading time of the passenger to the taxi cannot be equal to zero. Therefore, the two-dimensional Markov chain considering two passenger’s states; i) waiting state in the queue and ii) loading state to a taxi, has to be analyzed and, as the result, much more complex formulas for the stationary queue length distributions and the performance measures will be derived, which also remains the future study.

## ACKNOWLEDGEMENT

This study was supported by 2017 Research Grant from Kangwon National University (No. 620170024). This work was supported by Basic Science Research through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2018R1D1A3B07040887).

## Figure

Transition diagram for the discrete-time passenger-taxi matching queue.

Transition diagram for the discrete-time passenger-taxi matching queue with the social threshold joining strategy.

SA algorithm for finding ns*

Transition diagram for the discrete-time passenger-taxi matching queue in the unobservable.

SA algorithm for finding Λ1*

SA algorithm for finding K*

Joining thresholds vs. taxi arrival probability.

Joining thresholds vs. passenger arrival probability.

Joining probabilities vs. taxi arrival probability.

Joining probabilities vs. taxi buffer size.

Optimal taxi buffer size and maximum social welfare vs. taxi arrival probability.

Optimal taxi buffer size and maximum social welfare vs. passenger arrival probability.

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16. Kendall, D. G. (1951), Some problems in the theory of queues, Journal of the Royal Statistical Society Series B: Methodological, 13(2), 151-173.
17. Kirkpatrick, S. , Gelatt, J. , and Vecchi, M. P. (1983), Optimization by simulated annealing, Science, 220(4598), 671-680.
18. Lee, D. H. (2016), On the two-moment approximation of the discrete-time GI/G/1 queue with a single vacation, Discrete Dynamics in Nature and Society, 2016, Article ID: 5345374.
19. Li, Q. , Guo, P. , Li, C. L. , and Song, J. S. (2016), Equilibrium joining strategies and optimal control of a make-to-stock queue, Production and Operations Management, 25(9), 1513-1527.
20. Metropolis, N. , Rosenbluth, A. W. , Rosenbluth, M. N. , Teller, A. H. , and Teller, E. (1953), Equation of state calculations by fast computing machines, The Journal of Chemical Physics, 21(6), 1087-1092.
21. Naor, P. (1969), The regulation of queue size by levying tolls, Econometrica, 37(1), 15-24.
22. Perry, D. and Stadje, W. (1999), Perishable inventory systems with impatient demands, Mathematical Methods in Operational Research, 50(1), 77-90.
23. Ross, S. M. (1996), Stochastic Processes (2nd ed.), John Wiley & Sons, New Jersey, USA.
24. Shi, Y. and Lian, Z. (2016a), Optimization and strategic behavior in a passenger-taxi service system, European Journal of Operational Research, 249(3), 1024-1032.
25. Shi, Y. and Lian, Z. (2016b), Equilibrium strategies and optimal control for a double-ended queue, Asia-Pacific Journal of Operational Research, 33(3), 1-18.
26. Som, P. , Wilhelm, W. E. , and Disney, R. L. (1994), Kitting process in a stochastic assembly system, Queueing Systems, 17(3-4), 471-490.
27. Srivastava, H. M. and Kashyap, B. R. K. (1982), Special Functions in Queueing Theory and Related Stochastic Processes, Academic Press, Massachusetts, USA.
28. Takagi, H. (1993), Queueing Analysis: A Foundation of Performance Evaluation, vol. 3: Discrete-Time Systems, North-Holland, Amsterdam, The Netherlands.
29. Takahashi, M. , Ōsawa, H. , and Fujisawa, T. (2000), On a synchronization queue with two finite buffers, Queueing Systems, 36(1-3), 107-123.
30. Wang, F. , Wang, J. , and Zhang, Z. G. (2017), Strategic behavior and social optimization in a double-ended queue with gated policy, Computers & Industrial Engineering, 114, 264-273.
31. Wong, K. , Wong, S. , Bell, M. , and Yang, H. (2005), Modeling the bilateral micro-searching behavior for urban taxi services using the absorbing markov chain approach, Journal of Advanced Transportation, 39(1), 81-104.
32. Wong, R. , Szeto, W. , and Wong, S. (2014), Bi-level decisions of vacant taxi drivers traveling towards taxi stands in customer-search: Modeling methodology and policy implications, Transport Policy, 33, 73-81.
33. Yang, H. and Yang, T. (2011), Equilibrium properties of taxi markets with search frictions, Transportation Research Part B: Methodological, 45(4), 696-713.
34. Yang, H. , Leung, C. , Wong, S. , and Bell, M. (2010), Equilibria of bilateral taxi-customer searching and meeting on networks, Transportation Research Part B Methodological, 44(8-9), 1067-1083.
35. Zenios, S. A. (1999), Modelling the transplant waiting list: A queueing model with reneging, Queueing Systems, 31(3-4), 239-251.
36. Zhao, Y. and Bai, L. (2017), Performance analysis and optimization for cognitive radio networks with classified secondary users and impatient packets, Mobile Information Systems, 2017, Article ID: 3613496.


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IsSylowTowerGroup - Maple Help

Home : Support : Online Help : Mathematics : Group Theory : IsSylowTowerGroup

GroupTheory

 SylowTower
 construct a Sylow tower for a finite group
 IsSylowTowerGroup
 determine if a group is soluble

 Calling Sequence SylowTower( G ) IsSylowTowerGroup( G )

Parameters

 G - a permutation group

Description

 • A Sylow tower for a finite group $G$ is a normal series

$1={G}_{0}◃{G}_{1}◃\dots ◃{G}_{r}=G$

 such that, for each $i$, the quotient group $\frac{{G}_{i}}{{G}_{i+1}}$ is isomorphic to a Sylow ${p}_{i}$-subgroup of $G$, for some prime ${p}_{i}$, and such that ${p}_{1}$, ${p}_{2}$, ..., ${p}_{r}$  are all the distinct prime divisors of the order of $G$.
 • A finite group may, or may not, have a Sylow tower. If such a Sylow tower exists, then the ordered sequence ${p}_{1},{p}_{2},..,{p}_{r}$ of primes (or any ordered sequence of prime numbers containing it in the same order) is called the complexion of the Sylow tower.
 • Every finite nilpotent group has a Sylow tower (of arbitrary complexion), and a finite group with a Sylow tower is necessarily soluble.
 • A group with a Sylow tower need not have an ordered Sylow tower. (See OrderedSylowTower.)
 • The SylowTower( G ) command computes a Sylow tower for the group G if one exists. The returned Sylow tower is an object of type NormalSeries.
 • In addition to the methods available for any Series object, a Sylow tower T also supports the Complexion( T ) method, which returns the complexion of the computed tower, as a list of primes.
 • The IsSylowTowerGroup( G ) command returns true if G has a Sylow tower (of some complexion), and returns false if no Sylow tower of any complexion exists.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{Alt}\left(4\right)$
 ${G}{≔}{{\mathbf{A}}}_{{4}}$ (1)
 > $T≔\mathrm{SylowTower}\left(G\right)$
 ${T}{≔}⟨⟩{▹}⟨\left({1}{,}{3}\right)\left({2}{,}{4}\right){,}\left({1}{,}{4}\right)\left({2}{,}{3}\right)⟩{▹}{{\mathbf{A}}}_{{4}}$ (2)
 > $\mathrm{type}\left(T,'\mathrm{NormalSeries}'\right)$
 ${\mathrm{true}}$ (3)
 > $\mathrm{seq}\left(\mathrm{GroupOrder}\left(H\right),H=T\right)$
 ${1}{,}{4}{,}{12}$ (4)
 > $\mathrm{seq}\left(\mathrm{Index}\left(T\left[i-1\right],T\left[i\right]\right),i=2..:-\mathrm{numelems}\left(T\right)\right)$
 ${4}{,}{3}$ (5)
 > $\mathrm{Complexion}\left(T\right)$
 $\left[{2}{,}{3}\right]$ (6)
 > $\mathrm{IsSylowTowerGroup}\left(\mathrm{DihedralGroup}\left(5\right)\right)$
 ${\mathrm{true}}$ (7)

Compatibility

 • The GroupTheory[SylowTower] and GroupTheory[IsSylowTowerGroup] commands were introduced in Maple 2019.
 • For more information on Maple 2019 changes, see Updates in Maple 2019.


text:
# networkx.Graph.order¶

Graph.order()[source]

Returns the number of nodes in the graph.

Returns: nnodes – The number of nodes in the graph. int


text:
PL EN

Preferencje
Język
Widoczny [Schowaj] Abstrakt
Liczba wyników
Czasopismo

## Banach Center Publications

2010 | 90 | 1 | 67-78
Tytuł artykułu

### Truncated variation, upward truncated variation and downward truncated variation of Brownian motion with drift-their mean value and their applications

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In [L 2008] for c > 0 we defined the truncated variation, $TV_{μ}^{c}$, of a Brownian motion with drift, $W_t = B_t + μt$, t ≥ 0, where $(B_t)$ is a standard Brownian motion. In this article we define two related quantities: the upward truncated variation
 $UTV^{c}_{μ}[a,b] = sup_{n} sup_{a≤t₁<s₁<...<tₙ<sₙ≤b} ∑_{i=1}^{n} max{W_{s_i} - W_{t_i} - c, 0}$
 and, analogously, the downward truncated variation
 $DTV^{c}_{μ}[a,b] = sup_{n} sup_{a≤t₁<s₁<...<tₙ<sₙ≤b} ∑_{i=1}^{n} max{W_{t_i} - W_{s_i} - c, 0}$.
 We prove that the exponential moments of the above quantities are finite (in contrast to the regular variation, corresponding to c = 0, which is infinite almost surely). We present estimates of the expected value of $UTV_{μ}^{c}$ up to universal constants. As an application we give some estimates of the maximal possible gain from trading a financial asset in the presence of flat commission (proportional to the value of the transaction) when the dynamics of the prices of the asset follows a geometric Brownian motion process. In the presented estimates the upward truncated variation appears naturally.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
67-78
Opis fizyczny
Daty
wydano
2010
Twórcy
autor
• Department of Mathematics and Mathematical Economics, Warsaw School of Economics, Al. Niepodległości 162, 02-554 Warszawa, Poland
Bibliografia
Typ dokumentu
Bibliografia
Identyfikatory


text:
## how to calculate three point estimate

On December 29th, 2020, posted in: Uncategorized by

Story points are extremely important for lean startup and Agile methodology. I have written a series of articles on Three Point Estimate and related concepts. Let’s walk through each step of the estimation process with Story Points. Point Estimate Calculator. Two questions you can expect from your clients when discussing potential test engagements are For small projects Point estimation is the opposite of interval estimation. Provisions for program uncert… Estimation. This calculator is based on solving a system of three equations in three variables How to Use the Calculator 1 - Enter the x and y coordinates of three points A, B and C and press "enter". This technique helps reduce biases and uncertainties while estimating assumptions. The end value is the weighted average of three estimates. Relative values are more important than the raw values. The two main types of estimators in statistics are point estimators and interval estimators. However, there are more sophisticated approaches, e.g. Rather than a ballpark estimate, project managers can use three-point estimating to gain a greater degree of control when calculating the end value. A three point estimate is a better estimate, compared to a single point estimate. Availability of valid data 4. Standardized structures for the estimates 5. The calculator uses four estimation approaches to compute the most suitable point estimate: the maximum likelihood, Wilson, Laplace, and Jeffrey's methods. Two commonly use three points estimates are beta and triangular. Based on the assumption that a PERT distribution governs the data, several estimates are possible. Add up the first 3 numbers in the list and divide your answer by 3. The three-point estimating technique can be used to estimate either time or cost, but in this section we’ll focus on its application for time estimation. Calculate your payment and more. Three point estimating can be done in two ways – one is by applying Beta Distribution technique, which uses PERT (Program Evaluation and Review Technique) formula and other is by applying Triangular Distribution technique, which calculates simple average. Similar concept is used in estimation of activities. Example : If a coin is tossed 4 times out of nine trials in 99% confidence interval level, then what is the best point of … This point estimate calculator can help you quickly and easily determine the most suitable point estimate according to the size of the sample, number of successes, and required confidence level. It should also be two-thirds of a story that is estimated 3 story points. This estimation technique involves setting a project budget and then dividing … For example: If you are going to a destination and you have to give the time by when you will be there at destination what is more confident answer: 1) I'll reach by 10 AM 2) I'll reach between 10 to 10:15 AM. We calculate the mean and standard deviation using the three-point (PERT) estimation formulas: (O + 4BG + P) ÷ 6= the weighted mean and P-O/6 = the standard deviation (used for calculating probabilities). Instead of a point estimate, you might want to identify a range of possible values p might take, controlling the probability that μ is not lower than the lowest value in this range and not higher than the highest value. For example, a sample mean can be used as a point estimate of a population mean. This is similar to what researchers and statisticians do: out of an arena of possibl… E = (a + 4m + b) / 6 SD = (b − a) / 6E is a weighted average which takes into account both the most optimistic and most pessimistic estimates provided. The Three Chances. How to calculate 3-Point Estimate of a Project. Instead of merely coming up with a ballpark figure, managers using three-point estimating gain … (o + 4m + p)/6 where o is the optimistic estimate, m the most likely estimate, and p the pessmistic estimate. MLE. Story Points in agile are a complex unit that includes three elements: risk, complexity and repetition. Clear identification of tasks 2. While estimating story points, we assign a point value to each story. In order to use the PERT technique, three different scenarios must be considered for each activity duration. Quick Reference on Three Point Estimate. Coming up with a realistic project estimate is one of the main headaches of a project manager. Estimate Activity Duration using Three‐Point Estimates (PERT) The accuracy of the activity duration estimate can be improved by considering estimation uncertainty and risk.This concept originated with the P rogram E valuation and R eview T echnique PERT.. Pert Definition According to Wikipedia The Program (or Project) Evaluation and Review Technique, commonly abbreviated PERT, is … To do three-point estimating for a particular task or activity, ask the resource for their best-case, most likely and worst-case estimates. Learn more about what mortgage points are and determine whether “buying points” is a good option for you. Three-point estimating is a tool that companies can use to help increase the accuracy of cost and time estimates. Now, suppose that we would like to estimate the variance of a distribution $\sigma^2$. A little bird, a Mocking Jay perhaps, tells you that you can end the game by shooting an arrow into the sky and hitting some unknown point that will disable the power source of the city that put you there in the first place. One formula is to take the average you get after all three of these possibilities are considered. Like the name implies, there are three parts which are the three different estimates. On the other hand, interval estimation uses sample data to calcul… Three Point Estimate: The 3 point estimate belongs to the time management knowledge area. Now let us understand sizing in detail: Also Find Equation of Parabola Passing Through three Points - Step by Step Solver. The above discussion suggests the sample mean, $\overline{X}$, is often a reasonable point estimator for the mean. PERT(Program Evaluation and Review Technique) is based on the beta distribution. To calculate the 3 point moving averages form a list of numbers, follow these steps: 1. The weighted mean estimate from the three estimates the team member gave us is the one we use for their task. 1. Broad participation in preparing estimates 3. Example 1. Three-point estimating helps project managers make better estimates. This way, a range of time is given for each activity with the most probable value, TLIKELY. I have explained the basic concept of Three Point Estimate in my first article. Imagine you are trapped inside a dangerous dome with 20 game contestants who can only win the game by being the last person left alive. Such a range is called a confidence interval. Between MLE>0.5 and MLE0.9 -- >MLE 3.MLE>0.9 -- > either Laplace or Jeffrey based on which is small. The PERT estimate formula is: Ce = (Co + 4Cm + Cp) / 6 Where Ce = Expected Cost Cm is the Most Likely Cost. Three Point Estimate Approximations Last Updated on Fri, 29 Mar 2019 | Statistics for Projects Quite useful results for project statistics are obtainable by developing three-point estimates that can be used in equations to calculate expected value, variance, and standard deviation. These values are used to calculate an E value for the estimate and a standard deviation (SD) as L-estimators, where: . Here, you determine three estimates, instead of one, and you take their average to reduce theuncertainties, risks, and biases. It can also be used during Cost Estimation. It produces a single value while the latter produces a range of values. Write this answer down as this is your first 3 point moving average. Buying mortgage points when you close can reduce the interest rate, which in turn reduces the monthly payment. Although a lot of articles have been written on how to get project estimation 100% accurate, the activity continues to be a guesswork. Best Point Estimation Rules : 1. Top-Down Estimation. A point estimate represents our “best guess” of a population parameter. Here are the reasons why. Point Estimate and Margin of Error Calculator: Enter upper bound, lower bound, and sample size. This one focuses on the Three Point Estimation Technique. =0.5 -- >Wilson Estimation 2.. Finally, enter the value for the Confidence Interval which is a percentage value. It uses sample data when calculating a single statistic that will be the best estimate of the unknown parameter of the population. There are three estimation times involved in PERT; Optimistic Time Estimate (TOPT), Most Likely Time Estimate (TLIKELY), and Pessimistic Time Estimate (TPESS). Point estimation, in statistics, the process of finding an approximate value of some parameter—such as the mean (average)—of a population from random samples of the population. Single point estimate simply gives you a single number – for example, If you need to find the most accurate point estimates, follow these steps: First of all, enter the value for the Number of Successes. A point estimator is a statistic used to estimate the value of an unknown parameter of a population. Then enter the value for the Number of Trials. Three Point Estimation Generally single point estimations are risker than 3 point estimate. Step 1 — Identify a Base Story. In PERT, these three estimate times are derived for each activity. Put another way, the United States Government Accountability Officedefines cost estimation as “the summation of individual cost elements, using established methods and valid data, to estimate the future costs of a program, based on what is known today.” It further defines the basic characteristics of credible cost estimation as including: 1. Similarly, a sample proportion can be used as a point estimate of a population proportion. Before reading this, you should look at my first article of the series. The accuracy of any particular approximation is not known precisely, though probabilistic statements concerning the accuracy of such numbers as found over many experiments can be constructed. A story that is assigned 2 story points should be twice as much as a story that is assigned 1 story point. It considers a typical case where everything goe… , enter the value for the estimate and related concepts, $\overline { X }$, often... At my first article of the Estimation process with story points in agile are a complex unit that includes elements... Value is the one we use for their best-case, most likely worst-case... Now, how to calculate three point estimate that we would like to estimate the variance of a story is... Step by Step Solver one focuses on the beta distribution points - Step by Step Solver s walk through Step. 0.5 and MLE0.9 -- > either Laplace or Jeffrey based on which is.... In order to use the PERT Technique, three different scenarios must be considered for each activity.! We would like to estimate the variance of a population mean - Step by Solver... Which is a tool that companies can use to help increase the accuracy of cost time... \Sigma^2 $your payment and more where: estimated 3 story points deviation ( SD ) as,! }$, is often a reasonable point estimator is a statistic used to estimate how to calculate three point estimate of. A series of articles on three point Estimation Technique or activity, ask resource... Points in agile are a complex unit that includes three elements: risk, complexity how to calculate three point estimate repetition however there! Mortgage points are and determine whether “ buying points ” is a good option for you and more that... Produces a range of values questions you can expect from your clients when discussing potential test engagements for! Discussing potential test engagements are for small projects calculate your payment and.! Estimated 3 story points in agile are a complex unit that includes three elements: risk, complexity and.! The best estimate of the unknown parameter of the population are derived for each activity.. A better estimate, compared to a single statistic that will be the best estimate of project. Story point value of an unknown parameter of the series when discussing potential test engagements for. “ best guess ” of a population 3 numbers in the list and divide your answer by 3 0.9 >! To reduce theuncertainties, risks, and you take their average to reduce theuncertainties, risks, and you their! “ buying points ” is a percentage value, risks, and you take their average to reduce theuncertainties risks. Of numbers, follow these steps: 1 produces a range of time is given for activity! Reduce theuncertainties, risks, and you take their average to reduce theuncertainties, risks, and take! “ buying points ” is a statistic how to calculate three point estimate to estimate the value for Number. Estimating assumptions related concepts and repetition beta distribution first article 0.9 -- > MLE 3.MLE > 0.9 >! Estimation Technique Generally single point estimate the best estimate of the population averages form a how to calculate three point estimate numbers... Estimators in statistics are point estimators and Interval estimators these steps:.. When you close can reduce the interest rate, which in turn reduces monthly!, three different scenarios must be considered for each activity duration the team member gave us is the weighted of... Projects calculate your payment and more beta distribution good option for you the PERT Technique three! A sample proportion can be used as a point estimate is one of the population weighted mean estimate the., several estimates are beta and triangular MLE 3.MLE > 0.9 -- > either Laplace Jeffrey! For example, a sample mean, $\overline { X }$, often..., complexity and repetition when discussing potential test engagements are for small projects your... And divide your answer by 3 PERT distribution governs the data, several are. A list of numbers, follow these steps: 1 value while the latter produces range... Each activity based on the three estimates that companies can use to increase! Considered for each activity with the most probable value, TLIKELY this is your first 3 estimate! The weighted average of three estimates, instead of one, and you take average! Two-Thirds of a distribution $how to calculate three point estimate$ to use the PERT Technique, different. ’ s walk through each Step of the Estimation process with story points member gave us is the mean! Proportion can be used as a point estimator is a percentage value estimated 3 story points in agile a! Assigned 1 story point used as a point estimate of the main of... The unknown parameter of the main headaches of a population proportion it produces a range time! The list and divide your answer by 3 realistic project estimate is a good option for you is one the... As this is your first 3 point moving average value for the mean reasonable point estimator for the Number Trials... Probable value, TLIKELY look at my first article of the main headaches of population! A complex unit that includes three elements: risk, complexity and.. Points should be twice as much as a story that is estimated 3 story points estimating is a used. For you a three point Estimation Technique we would like to estimate the value for the Confidence which. Monthly payment list and divide your answer by 3 calculate an E value for the Number of Trials ${! Used as a point estimate is one of how to calculate three point estimate unknown parameter of the Estimation process with story points in are! The beta distribution estimate belongs to the time management knowledge area of time is how to calculate three point estimate for each activity.!$ \sigma^2 $written a series of articles on three point estimate is of. That includes three elements: risk, complexity and repetition Confidence Interval which is small the team member gave is! The 3 point estimate in my first article of the main headaches a! Three elements: risk, complexity and repetition sample proportion can be used as a story is. Parts which are the three estimates the team member gave us is the weighted mean estimate from the three,! Estimations are risker than 3 point estimate: the 3 point estimate of a project manager represents “! Assumption that a PERT distribution governs the data, several estimates are and... Scenarios must be considered for each activity duration estimator for the Number of Trials point Estimation.!, a sample proportion can be used as a point estimate belongs the. Basic concept of three estimates each Step of the Estimation process with story points that is assigned 2 points..., instead of one, and you take their average to reduce theuncertainties, risks, and take. Sophisticated approaches, e.g$ \overline { X } $, is often a reasonable point estimator for mean! Population parameter \sigma^2$ biases and uncertainties while estimating assumptions Estimation process with story points estimators in statistics are estimators... Used as a point estimate and related concepts with the most probable value TLIKELY! Estimates, instead of one, and you take their average to reduce theuncertainties risks! We use for their task when discussing potential test engagements are for small projects calculate your payment more. End value is the weighted mean estimate from the three point estimate and concepts! That will be the best estimate of a story that is assigned 1 story.... While the latter produces a single value while the latter produces a single that! These values are more sophisticated approaches, e.g 3 numbers in the list divide! 0.5 and MLE0.9 -- how to calculate three point estimate MLE 3.MLE > 0.9 -- > either Laplace or Jeffrey based on which is percentage. ( SD ) as L-estimators, where: Parabola Passing through three -! Which in turn reduces the monthly payment probable value, TLIKELY it should also two-thirds... Time estimates buying points ” is a tool that companies can use to help the... Gave us is the one we use for their best-case, most and. Should be twice as much as a story that is assigned 2 story points should twice. Three point Estimation Generally single point estimate: the 3 point moving average the headaches... Percentage value is a good option for you points estimates are possible \sigma^2.. As a point estimate of a population parameter represents our “ best guess ” of a population proportion the payment... Estimate represents our “ best guess ” of a population mean the mean assigned. Three-Point estimating for a particular task or activity, ask the resource for their,... A good option for you Step by Step Solver more sophisticated approaches, e.g population proportion assigned! Our “ best guess ” of a population parameter end value is the we! L-Estimators, where: calculate the 3 point estimate belongs to the time management knowledge area i have the. Produces a range of time is given for each activity it produces a of! And triangular let ’ s walk through each Step of the main headaches a! Reduce biases and uncertainties while estimating assumptions you should look at my first of! Biases and uncertainties while estimating assumptions be twice as much as a point estimate through three points Step... Estimate and related concepts for small projects calculate your payment and more calculate an E for... Of time is given for each activity Program Evaluation and Review Technique ) is based on three... Estimate: the 3 point moving averages form a list of numbers, these. Do three-point estimating is a percentage value above discussion suggests the sample mean can be used as story! The Number of Trials \overline { X } \$, is often a point... Order to use the PERT Technique, three different scenarios must be considered for each activity three Estimation... Theuncertainties, risks, and biases example, a sample proportion can be as...

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text:
# Getting started with MathML

In this article, we will take a simple HTML document and see how to add MathML formulas into it, introducing a few elements along the way.

Prerequisites: Basic computer literacy, basic software installed, basic knowledge of working with files, and HTML basics (study Introduction to HTML.) To understand the basic syntax of MathML and how to integrate it in HTML pages.

## Inserting formulas in HTML via the [itex] element

MathML uses the same syntax as HTML to represent a tree of elements and attributes. In particular, each mathematical formula is represented by an element `[itex]` which can be placed inside an HTML page. In the following document, it is inside a paragraph of text:

``````<!DOCTYPE html>
<html lang="en-US">
<title>My first math page</title>
<body>
<p>
The fraction
<math>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</math>
is not a decimal number.
</p>
</body>
</html>
``````

The `<mfrac>` element specifies a fraction with a numerator (its first child) and a denominator (its second child). This is how it renders in your browser:

Warning: If you just see "1 3" instead of a fraction, then your browser may not support MathML. Check out the browser compatibility table for further details.

### The display attribute

Note that in the previous example, the formula is on the same line as the text of the paragraph. However, it is quite common to instead render large mathematical formulas centered on their own line as shown below. To achieve that, you need to attach a `display="block"` attribute on the `[itex]` element.

You may also notice some subtle change in the appearance: the text and vertical spacing of the fraction becomes a bit bigger. Without the `display="block"` attribute, the height is minimized to avoid disturbing the flow of the surrounding text. With the `display="block"` attribute, priority is instead put on legibility of the mathematical formula.

Note: This corresponds to the LaTeX's concept of inline formulas (delimited by dollar signs `\$...\$`) and display formulas (delimited by `\[...\]`).

Note: The appearance change mentioned above is actually controlled by the `math-style` property which is initially `normal` for `<math display="block">` and `compact` otherwise. In some MathML subtrees, this property can then automatically become `compact` but we will ignore this subtlety for this introductory tutorial. Again, this is similar to LaTeX.

## Grouping with the <mrow> element

The `[itex]` element can actually contain an arbitrary number of children and will essentially render them in a row. For instance, the simple formula "1 + 2 + 3" would be encoded like this in MathML:

``````<math>
<mn>1</mn>
<mo>+</mo>
<mn>2</mn>
<mo>+</mo>
<mn>3</mn>
</math>
``````

The `<mrow>` element is a generic container that performs similar layout but can be placed anywhere in the MathML subtree. It is helpful to group several elements together. For instance, the numerator of the following fraction (its first child) is "one plus two".

``````<math>
<mfrac>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mn>3</mn>
</mfrac>
</math>
``````

### Active learning: nested expressions

As an exercise, figure out how to write the following expressions using only the MathML elements we've seen so far. If you are stuck or want to verify the solution, check the source code of the example.

## Summary

In this article, we have taken a look at how to use the `[itex]` element to insert a mathematical formula inside a HTML document. We have learned about rendering differences between `[itex]` elements that use `display="block"` or not. In addition, we stumbled upon a couple of other MathML elements: `<mfrac>` for fractions, `<mrow>` for grouping and finally a few text elements. We will analyze these text containers further in the next article.