Queueing Analysis and Nash Equilibria in an Unobservable

Taxi-passenger System with Two Types of Passenger

Hung Q. Nguyen

and Tuan Phung-Duc

2 a

Graduate School of Science and Technology, University of Tsukuba,

1-1-1 Tennodai, Tsukuba, Ibaraki 305-8573, Japan

Faculty of Engineering, Information and Systems, University of Tsukuba,

1-1-1 Tennodai, Tsukuba, Ibaraki 305-8573, Japan

Keywords:

Queueing Theory, Markov Chain, Game Theory, Strategic Queueing, Taxi-passenger System.

Abstract:

This paper considers an unobservable double-ended queueing system motivated by the application of a taxi

station where passengers and taxis arrive at two sides of the queue, and there are two types of passengers,

differentiated by their mean matching times with taxis. We use a three-dimensional Markov chain to model

the system and derive several system performance measures (mean queue lengths, waiting times and social

welfare). Furthermore, when agents are strategic, we model the system as a multi-population game among

three populations of agents and ﬁnd their joining rates in equilibrium.

1 INTRODUCTION

A taxi-passenger system is a typical real-life example

of a double-ended queueing system in which agents

arrive at both sides of the queue for matching. In air-

ports and railway stations, it is common to see both

passengers and taxis form queues to be matched: a

queue of passengers who wait for taxis, and a queue

of taxis that wait for passengers. This type of sys-

tem was ﬁrst studied by Kendall (1951), followed by

Dobbie (1961) and Di Crescenzo et al. (2012). Those

early studies did not incorporate strategic agent be-

haviors into the model.

A queueing analysis considering strategic cus-

tomer behavior was ﬁrst undertaken by Naor (1969).

This research identiﬁed a queue length threshold that

determines customers’ decision between joining or

balking, under the most basic setting of an observ-

able M/M/1 queue. Edelson and Hildebrand (1975)

complemented by considering the unobservable case.

Multiple extensions followed, comprehensively sum-

marized in Hassin and Haviv (2003); Hassin (2016).

Hassin and Haviv (2002); Economou (2021) provided

a concrete theoretical framework for this type of prob-

lem by deﬁning the problem under the scope of game

theory. However, only one-population games were

considered in those studies.

https://orcid.org/0000-0002-5002-4946

Combining the two aforementioned concepts re-

sults in a new topic—the strategic behavior of agents

in double-ended queueing systems, with several no-

table studies: Shi and Lian (2016) considered a sys-

tem where matching times are zero; hence, the system

is described by a single number which is the differ-

ence between the number of passengers and the num-

ber of taxis in the system. Wang et al. (2017) consid-

ered a system with a gated policy. Jiang et al. (2020)

incorporated customer loss aversion into the model.

Wang and Liu (2019) discussed the impacts of differ-

ent levels of information. In all of these studies, only

passengers are assumed to be rational.

In reality, due to bulky luggage or to communica-

tion between passengers and taxi drivers, the match-

ing process usually takes more time than can be dis-

missed as negligible. Furthermore, there are access

points where multiple passenger–taxi pairs can match

at the same time. Another consideration is that there

may be multiple types of passengers whose match-

ing times are not identical. For example, domestic

passengers can match with local taxi drivers more

quickly, while it may take a longer time for foreign

visitors to communicate with taxis drivers. Moreover,

agents from the supply side may also be strategic.

Motivated by these complex circumstances, this

paper analyzes a taxi-passenger and multi-server

queueing system, and studies agents’ strategies for

joining the queue based on individual utility. The

Nguyen, H. and Phung-Duc, T.

Queueing Analysis and Nash Equilibria in an Unobservable Taxi-passenger System with Two Types of Passenger.

DOI: 10.5220/0010825200003117

In Proceedings of the 11th International Conference on Operations Research and Enterprise Systems (ICORES 2022), pages 48-55

ISBN: 978-989-758-548-7; ISSN: 2184-4372

main contributions of this paper are:

• We consider a system with two passenger types,

distinguished by their differing service rates.

• We consider general non-zero matching times

which follow exponential distributions, and mul-

tiple access points.

• From the modeling perspective, we use a three-

dimensional Markovian description of the system.

When agents are strategic, the system is modeled

as a three-population game.

The remainder of this paper is structured as follows:

Section 2 describes the queueing system and presents

notations. Next, we derive some performance mea-

sures in Section 3. In Section 4, we derive Nash equi-

libria of the system when agents are strategic. In Sec-

tion 5, we analyze the system through several numer-

ical examples. Finally, Section 6 concludes the paper.

2 PRELIMINARIES

We consider an unobservable double-ended queueing

system with a taxi stand containing S identical access

points. There are three populations of agents arriving

at the system: type-1 passengers, type-2 passengers,

and taxis. The area (including S taxi access points)

can accommodate at most K taxis at the same time

(K ≥ S). In an ideal situation where agents are given

enough incentive to join the queue without balking,

passengers and taxis arrive at the taxi stands accord-

ing to Poisson processes with potential arrival rates

and Λ

, respectively. An access point receives

a type-1 passenger with probability ε, and a type-

2 passenger with probability 1 − ε. The matching

times of type-1 and type-2 passengers follow expo-

nential distributions with parameters µ

and µ

, re-

spectively. Without loss of generality, we can assume

that µ

< µ

, meaning that type-1 passengers have a

larger mean matching time. When the parking area

reaches its maximum capacity, the arrival of any new

taxi is blocked, so that taxi leaves immediately. On

the other hand, we assume that there is no limit on

the buffer of passengers. If a passenger arrives when

all access points are busy, or when there are no taxis

available for matching, he will wait in the queue under

the ﬁrst-come, ﬁrst-served (FCFS) service principle.

Let R

and R

denote the service values, and C

and C

denote the waiting cost per time unit of passen-

gers and taxis, respectively. Let q

, q

and q

denote

the joining probabilities of type-1 passengers, type-2

passengers, and taxis, respectively. Let λ

= (q

ε +

(1−ε))Λ

, λ

= q

and α =

ε+q

(1−ε)

. Then,

and λ

are the actual arrival rates of agents; these

will be are used to derive performance measures in

the following section.

3 PERFORMANCE MEASURES

Let L(t), I(t) and J(t) respectively denote the num-

ber of type-1 passengers being matched at the access

points, the number of taxis in the system, and the to-

tal number of passengers in the system, at time t. The

process {(L(t), I(t), J(t)) | t ≥ 0} is a continuous-time

Markov Chain with the state space S given by

S = {(l, i, j) ∈ {0, 1, ..., S} × {0, 1, ..., K} × {0, 1, 2, . . . }}.

Also, as implied by their deﬁnitions, it should be

noted that l ≤ i and l ≤ j.

The system can be modeled as a quasi-birth-death

process with the inﬁnitesimal generator Q being ex-

pressed as follows.

Q =







(0)

O O O ... ... ... ...

(1)

O O ...

O A

(2)

O ...

O O ... A

(K)

O ...

O O ... O A

(K)

O ...







, (1)

where O denotes a zero matrix of appropriate dimen-

sion. If we denote by M(m, n) the set of all m × n-

dimensional matrices, and M (i, j) (i, j ∈ Z

) the el-

ement at the i

row, j

column of a matrix M , then

the block matrices in Q can be deﬁned by the sys-

tem parameters λ

, λ

, µ

and α, as shown in the

Appendix.

Letting Q

∗

= A

(K)

, we can then de-

rive the stability condition of the system by simulta-

neously solving the following equations for η.

ηQ

∗

= 0,

and

ηe = 1,

where η is the row vector representing the stationary

distribution of the inﬁnitesimal generator Q

∗

, 0 is a

row vector with all elements equal to 0 and e is a col-

umn vector with all elements equal to 1. The stability

condition, then, is

ηC

(K)

e < ηA

(K)

e. (2)

If stability condition (2) is not satisﬁed, the sys-

tem becomes an M/H

/S/K queue of taxis in which

the passengers become “servers”. The mean waiting

Queueing Analysis and Nash Equilibria in an Unobservable Taxi-passenger System with Two Types of Passenger

times of two type-1 and type-2 passengers are given

= W

= +∞. (3)

When the stability condition holds, we can de-

rive the steady state probabilities π = (π

, π

, ...),

where π

= (π

0,0, j

, π

1,0, j

, ..., π

S,K, j

) is the vector en-

coding all probabilities when there are j passengers

in the system at the steady state. For j ≥ K, there

exists a constant matrix R such as

= π

j−K

where R satisﬁes

(K)

+ R B

(K)

+ R

(K)

= O. (4)

The solution of the matrix equation (4) is obtained

by the Matrix Geometric Method proposed by Neuts

(1981). Furthermore, for 1 ≤ j ≤ K, we also have

= π

j−1

( j)

where R

(K)

= R , and

(1)

, R

(2)

, ..., R

(K−1)

;π

, π

, ..., π

are recursively

calculated as

( j)

= −C

( j)

+ R

( j+1)

)

−1

is determined by solving

(0)

+ R

(1)

)

−1

= 0, (5)

and

I +

S−1

∑

j=1

∏

i=1

(i)

∏

i=1

(i)

(I − R )

−1

e = 1,

(6)

where I denotes an identity matrix of appropriate di-

mension.

Next, we derive performance measures of the sys-

tem, including average queue lengths, and average

waiting times of passengers and taxis. The average

number of passengers in the waiting line is given by

∞

∑

j=0

∑

i=0

∑

l=0

( j − min{i, j, S})π

l,i, j

K−1

∑

j=0

jπ

−

K−1

∑

j=0

+ π

(I − R )

−1

[KI + (I − R )

−1

R ]e

− π

(I − R )

−1

, (7)

where

• e

is a unit vector of the same dimension as π

• g

= (g

0,0, j

, g

1,0, j

, ..., g

S,K, j

), where g

l,i, j

min{i, j, S}.

Note that the summation in (7) excludes (l, i, j) not

existing in the state space. The same rule applies to

all later summations and products.

The average number of taxis in the system is given

∞

∑

j=0

∑

i=0

∑

l=0

iπ

l,i, j

K−1

∑

j=0

+ π

(I − R )

−1

. (8)

Here, f

= ( f

0,0, j

, f

1,0, j

, ..., f

S,K, j

), where f

l,i, j

= i.

Corresponding to (7) and (8), the average sojourn

time taxis can be calculated via Little’s Law as fol-

lows

(1 − P

)

, (9)

where P

is the blocking probability of taxis, calcu-

lated by

K−1

∑

j=0

+ π

(I − R )

−1

. (10)

Here, u

is a vector with the same dimension as π

, in

which the last min( j + 1, S + 1) elements equal 1 and

the other elements equal 0.

The average number of passengers in the system

is given by

∞

∑

j=0

jπ

K−1

∑

j=0

jπ

+ π

(I − R )

−1

[KI + (I − R )

−1

R ]e

(11)

The average sojourn time of all passengers is

given by

. (12)

Since a passenger knows his own type before en-

tering the taxi stand, the expected sojourn times of a

type-1 and a type-2 passenger are estimated as

, (13)

and

. (14)

Finally, social welfare, which equals the total util-

ity of all agents in the system per time unit, is given

SW = λ

+ λ

(1 − P

−C

. (15)

ICORES 2022 - 11th International Conference on Operations Research and Enterprise Systems

For further analysis, we acknowledge the follow-

ing properties, which are supported by intuition and

numerous simulation results.

Axiom 1. The expected sojourn time of an arbitrary

agent depends on the strategies of agents in their own

population and the other populations. In other words,

and W

are functions of q

, q

and q

. The

monotonic properties of these functions with respect

to each variable are given as follows.

Under the stability condition given in (2),

(1) W

is continuously increasing in q

and q

, and

decreasing in q

(2) W

is continuously increasing in q

and q

, and

decreasing in q

(3) W

is continuously decreasing in q

and q

, and

increasing in q

4 NASH EQUILIBRIA

In this section, we derive all possible Nash equilibria

at which agents make a best response to the strategies

of other agents. The social proﬁle, which is repre-

sented by a triplet (q

, q

), is denoted X. Let

X = ( ¯q

, ¯q

) be the social proﬁle in equilibrium.

Denote by U





the payoff of an arbitrary

type-1 passenger who adopts a strategy q

against the

social proﬁle

X, which is given by





= q



−C





. (16)

By deﬁnition, ¯q

is a best response against the

social proﬁle in equilibrium, which means

¯q

∈ argmax















{0} if R

−C





< 0,

[0, 1] if R

−C





= 0,

{1} if R

−C





> 0.

(17)

If we similarly deﬁne U





and U





for the other two populations, we also have

¯q

∈ argmax















{0} if R

−C





< 0,

[0, 1] if R

−C





= 0,

{1} if R

−C





> 0,

(18)

and

¯q

∈ argmax















{0} if R

−C





< 0,

[0, 1] if R

−C





= 0,

{1} if R

−C





> 0.

(19)

(17), (18) and (19) lead to 27 possible combinations.

However, we can reduce the number of cases to con-

sider by noting that W





> W





, and consider-

ing the following special cases. First, if ¯q

= 0, mean-

ing that taxis do not join the system, then it is easily

seen that ¯q

= ¯q

= 0 since the best response of pas-

sengers is not to join the system either. On the other

hand, if ¯q

= 0, meaning that type-2 passengers have

no incentive to join the system, then it is implied that

¯q

= 0 (since type-1 passengers always expect longer

sojourn times than type-2 passengers), which leads to

¯q

= 0. In other words, ( ¯q

, ¯q

) = (0,0, 0) is an

equilibrium and is the only equilibrium where ¯q

= 0

or ¯q

= 0. When ¯q

> 0, ¯q

> 0 and the stability con-

dition (2) is satisﬁed, all possible equilibria can be

derived as shown in Table 1.

Table 1: Equilibria and corresponding conditions.

Equilibria Conditions

(1, 1, 1)

−C

(1, 1, 1) > 0,

−C

(1, 1, 1) > 0,

−C

(1, 1, 1) > 0.

( ¯q

, 1, 1)

−C

( ¯q

, 1, 1) = 0,

−C

( ¯q

, 1, 1) > 0,

−C

( ¯q

, 1, 1) > 0.

(0, 1, 1)

−C

(0, 1, 1) < 0,

−C

(0, 1, 1) > 0,

−C

(0, 1, 1) > 0.

(0, ¯q

, 1)

−C

(0, ¯q

, 1) < 0,

−C

(0, ¯q

, 1) = 0,

−C

(0, ¯q

, 1) > 0.

(1, 1, ¯q

)

−C

(1, 1, ¯q

) > 0,

−C

(1, 1, ¯q

) > 0,

−C

(1, 1, ¯q

) = 0.

( ¯q

, 1, ¯q

)

−C

( ¯q

, 1, ¯q

) = 0,

−C

( ¯q

, 1, ¯q

) > 0,

−C

( ¯q

, 1, ¯q

) = 0.

(0, 1, ¯q

)

−C

(0, 1, ¯q

) < 0,

−C

(0, 1, ¯q

) > 0,

−C

(0, 1, ¯q

) = 0.

(0, ¯q

, ¯q

)

−C

(0, ¯q

, ¯q

) < 0,

−C

(0, ¯q

, ¯q

) = 0,

−C

(0, ¯q

, ¯q

) = 0.

Equilibria #1 and #3 can be veriﬁed by simply

checking their corresponding conditions. The other

equilibria are derived by solving their corresponding

conditional equations and double-checking other con-

ditions. The solutions to those equations are not ex-

plicit but are computationally solvable. We will il-

lustrate results in several numerical examples in the

following section.

Queueing Analysis and Nash Equilibria in an Unobservable Taxi-passenger System with Two Types of Passenger

5 NUMERICAL ANALYSIS

In this section, we present the analysis in speciﬁc nu-

merical examples. First, we assume that agents are

not rational, and numerically examine the variation of

some performance measures with respect to system

parameters and actual joining rates. In the following

examples, we set µ

= 2, µ

= 5, α = 0.3, R

= 15,

= 20, C

= 5 and C

= 4. Results are illustrated in

Figs. 1 to 8.

Figs. 1 to 4 verify the monotonic properties of pas-

sengers’ and taxis’ waiting times with respect to the

agents’ actual joining rates. The results are intuitive

and follow exactly as in Axiom 1.

Figure 1: W

w.r.t λ

(λ

= 5, K = 20).

Figure 2: W

w.r.t λ

(λ

= 4, K = 20).

Figs. 5 and 6 show that the social welfare func-

tion is unimodal with respect to the joining rates of

passengers and taxis. There exists a value of passen-

gers’ (or taxis’) joining rate at which social welfare

reaches its maximum. These patterns suggest that the

platform manager can control the arrival rate of one

side of agents in case the other side is not strategic,

to maximize social welfare. More details about appli-

cable control policies can be found in Haviv and Oz

(2018). For example, when taxi drivers are not strate-

gic and join the queue with rate λ

= 5, policy makers

can interfere in the passengers’ service value to ad-

just their arrival rate at around λ

∗

= 4.7, which yields

Figure 3: W

w.r.t λ

(λ

= 5, K = 20).

Figure 4: W

w.r.t λ

(λ

= 4, K = 20).

the highest social welfare. When λ

< λ

∗

, passengers

need more incentive to join the queue, so a ﬁxed sub-

sidy (such as a discount or coupon) can be granted.

On the contrary, when λ

> λ

∗

, a toll fee can be ap-

plied to reduce the joining rate of passengers. The

same policies apply in the case where taxi drivers are

strategic and passengers are not.

Figure 5: SW w.r.t λ

(λ

= 5, S = 3, K = 20).

Figs. 7 and 8 illustrate how social welfare varies

according to the two system parameters S and K. In

this experiment, social welfare increases quickly at

ﬁrst, then remains almost unchanged as S becomes

larger. It can be observed that 5 access points are

enough and optimal in this example (considering that

ICORES 2022 - 11th International Conference on Operations Research and Enterprise Systems

Figure 6: SW w.r.t λ

(λ

= 4, S = 3, K = 20).

a larger parking lot may consume more budget for

construction and management). On the contrary, so-

cial welfare decreases with increased parking capac-

ity. This phenomenon may stem from the fact that the

parking capacity already exceeds a particular “thresh-

old,” above which the queue length of taxis gets

longer and leads to inefﬁcient waiting times, thus re-

ducing social welfare.

Figure 7: SW w.r.t S (λ

= 4, λ

= 5, K = 20).

Figure 8: SW w.r.t K (λ

= 4, λ

= 5, S = 3).

In what follows, we derive joining probabilities of

agents and calculate social welfare in the case where

agents are strategic. For this, we set Λ

= 4, Λ

= 5,

= 1, µ

= 5, S = 3, K = 8, ε = 0.3, C

= 5 and

= 4.

First, it can be seen that the equilibrium (0, 0, 0)

exists in any setting of parameters. In reality, the

system may end up at the equilibrium (0, 0, 0) in ex-

treme situations, for example, when the system is ter-

minated. In the following example, we derive other

equilibria from the situations in Table 1.

Example 1. Assume R

= 15 and R

= 20. In this

case, two equilibria exist: (1, 1, 1) (and (0, 0, 0)). This

means either that potential agents all join, or that none

join at all.

Example 2. Assume R

= 15 and R

= 4. In this ex-

ample, we keep the service value of passengers un-

changed while reducing the service value of taxis.

This make the equilibrium (1, 1, 1) no longer exist

since taxis expect a negative payoff when they join

the system at full rate. Instead, there exists an equilib-

rium at (0, 0.9099, 0.5499), at which type-1 passen-

gers choose not to join the system at all, while both

type-2 passengers and taxis join the system at a rate

smaller than the corresponding potential rate.

Example 3. Assume R

= 2.5 and R

= 4. In this

example, we found the equilibrium (0, 1, 1), meaning

that type-1 passengers choose not to join the system

at all, while both type-2 passengers and taxis join the

system at full potential rates.

6 CONCLUSIONS

This paper examined the variations in social welfare

and waiting times of agents in a taxi-passenger system

with respect to changes in joining rates and system

parameters. The derivation of such performance mea-

sures provided a basis for further optimization of the

system and the identiﬁcation of equilibrium joining

rates when agents are strategic. We derived different

patterns of Nash equilibria and showed that multiple

equilibria may simultaneously exist in speciﬁc numer-

ical examples.

Future work may consider a solution for the so-

cial welfare optimization problem on three decision

variables corresponding to the joining probabilities of

three populations of agents. The results of such inves-

tigations provides for the proposal of socially optimal

policies.

ACKNOWLEDGEMENTS

The research of Hung Q. Nguyen is supported by

JST SPRING, Grant Number JPMJSP2124. The re-

search of Tuan Phung-Duc is supported in part by

JSPS KAKENHI Grant Number 21K11765.

Queueing Analysis and Nash Equilibria in an Unobservable Taxi-passenger System with Two Types of Passenger

REFERENCES

Di Crescenzo, A., Giorno, V., Kumar, B. K., and Nobile,

A. G. (2012). A double-ended queue with catastro-

phes and repairs, and a jump-diffusion approximation.

Methodology and Computing in Applied Probability,

14(4):937–954.

Dobbie, J. M. (1961). Letter to the editor—a doubled-ended

queuing problem of Kendall. Operations Research,

9(5):755–757.

Economou, A. (2021). The Impact of Information Structure

on Strategic Behavior in Queueing Systems, pages

137–169. Wiley-ISTE.

Edelson, N. M. and Hildebrand, K. (1975). Congestion

tolls for Poisson queueing processes. Econometrica,

43(1):81–92.

Hassin, R. (2016). Rational queueing. Chapman and

Hall/CRC, New York, 1 edition.

Hassin, R. and Haviv, M. (2002). Nash equilibrium and

subgame perfection in observable queues. Annals of

Operations Research, 113:15–26.

Hassin, R. and Haviv, M. (2003). To queue or not to queue:

Equilibrium behavior in queueing systems. Kluwer

Academic Publishers, Boston.

Haviv, M. and Oz, B. (2018). Self-regulation of an unob-

servable queue. Management Science, 64(5):2380–

2389.

Jiang, T., Chai, X., Liu, L., Lv, J., and Ammar, S. I.

(2020). Optimal pricing and service capacity manage-

ment for a matching queue problem with loss-averse

customers. Optimization.

Kendall, D. G. (1951). Some problems in the theory of

queues. Journal of the Royal Statistical Society: Se-

ries B (Methodological), 13(2):151–173.

Naor, P. (1969). The regulation of queue size by levying

tolls. Econometrica, 37(1):15–24.

Neuts, M. F. (1981). Matrix geometric solutions in stochas-

tic models - An algorithmic approach. Johns Hopkins

University Press, Baltimore, 1 edition.

Shi, Y. and Lian, Z. (2016). Optimization and strategic be-

havior in a passenger-taxi service system. European

Journal of Operational Research, 249(3):1024–1032.

Wang, F., Wang, J., and Zhang, Z. G. (2017). Strategic

behavior and social optimization in a double-ended

queue with gated policy. Computers & Industrial En-

gineering, 114:264–273.

Wang, Y. and Liu, Z. (2019). Equilibrium and optimiza-

tion in a double-ended queueing system with dynamic

control. Journal of Advanced Transportation, 2019.

APPENDIX

(K)

, B

(K)

, C

(K)

∈ M



(S + 1)(S + 2)

+ (K − S)(S + 1),

(S + 1)(S + 2)

+ (K − S)(S + 1)



such that

(K)

= diag(λ, λ, ..., λ);

(K)



(S + 1)(S + 2)

+ i(S + 1) + j,

S(S + 1)

+ i(S + 1)

+ j



= α( j − 1)µ

+ (1 − α)(S − ( j − 1))µ

(K)



(S + 1)(S + 2)

+ i(S + 1) + j,

S(S + 1)

+ i(S + 1) + ( j + 1)



= α(S − ( j − 1))µ

(K)



(S + 1)(S + 2)

+ i(S + 1)

+ ( j + 1),

S(S + 1)

+ i(S + 1) + j



= (1 − α) jµ

(K)



S(S + 1)

+ i(S + 1) + j,

(S + 1)(S + 2)

+ i(S + 1) + ( j + 1)



= λ

for i = 0, 1, ..., K − S and j = 1, 2, ..., S + 1;

(K)



(i + 1)(i + 2)

+ j,

i(i + 1)

+ j



= (i − ( j − 1))µ

(K)



(i + 1)(i + 2)

+ ( j + 1),

i(i + 1)

+ j



= jµ

(K)



i(i + 1)

+ j,

(i + 1)(i + 2)

+ j



= (1 − α)λ

(K)



i(i + 1)

+ j,

(i + 1)(i + 2)

+ ( j + 1)



= αλ

for i = 0, 1, ..., S −1 and j = 1, 2, ..., i + 1.

For n < K,

(n)

∈ M



(n + 1)(n + 2)

+ (K − n)(n + 1),

(n + 1)(n + 2)

+ (K − n)(n + 2)



for n ≥ 1, and

ICORES 2022 - 11th International Conference on Operations Research and Enterprise Systems

(n)

∈ M



(n + 1)(n + 2)

+ (K − n)(n + 1),

n(n + 1)

+ (K − (n − 1))n



(n)

∈ M



(n + 1)(n + 2)

+ (K − n)(n + 1),

(n + 1)(n + 2)

+ (K − n)(n + 1)



such that

(n)

(i, i) = λ

for i = 1, 2, ...,

(n+1)(n+2)

;

(n)



(n + 1)(n + 2)

+ i(n + 1)

+ j,

(n + 1)(n + 2)

+ i(n + 2) + j



= (1 − α)λ,

(n)



(n + 1)(n + 2)

+ i(n + 1)

+ j,

(n + 1)(n + 2)

+ i(n + 2) + ( j + 1)



= αλ,

(n)



(n + 1)(n + 2)

+ i(n + 1) + j,

n(n + 1)

+ i(n + 1) + j



= (n − ( j − 1))µ

(n)



(n + 1)(n + 2)

+ i(n + 1)

+ ( j + 1),

n(n + 1)

+ i(n + 1) + j



= jµ

(n)



n(n + 1)

+ i(n + 1) + j,

(n + 1)(n + 2)

+ i(n + 1) + ( j + 1)



= λ

for i = 0, 1, ..., K − n and j = 1, 2, ..., n + 1;

(n)



(i + 1)(i + 2)

+ j,

i(i + 1)

+ j



= (i − ( j − 1))µ

(n)



(i + 1)(i + 2)

+ ( j + 1),

i(i + 1)

+ j



= jµ

(n)



i(i + 1)

+ j,

(i + 1)(i + 2)

+ j



= (1 − α)λ

(n)



i(i + 1)

+ j,

(i + 1)(i + 2)

+ ( j + 1)



= αλ

for i = 0, 1, ..., n − 1 and j = 1, 2, ..., i + 1; and n ≥ 1.

Finally,

(0)

(i, i) = −

∑

(0)

(i, j) −

∑

j6=i

(0)

(i, j),

and

(n)

(i, i) = −

∑



(n)

(i, j) +C

(n)

(i, j)



−

∑

j6=i

(n)

(i, j)

for n = 1, ..., K.

Queueing Analysis and Nash Equilibria in an Unobservable Taxi-passenger System with Two Types of Passenger