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
Λ
p
and Λ
t
, 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
1
and µ
2
, re-
spectively. Without loss of generality, we can assume
that µ
1
< µ
2
, 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
p
and R
t
denote the service values, and C
p
and C
t
denote the waiting cost per time unit of passen-
gers and taxis, respectively. Let q
p
1
, q
p
2
and q
t
denote
the joining probabilities of type-1 passengers, type-2
passengers, and taxis, respectively. Let λ
p
= (q
p
1
ε +
q
p
2
(1−ε))Λ
p
, λ
t
= q
t
Λ
t
and α =
q
p
1
ε
q
p
1
ε+q
p
2
(1−ε)
. Then,
λ
p
and λ
t
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 =
B
(0)
C
(0)
O O O ... ... ... ...
A
(1)
B
(1)
C
(1)
O O ...
.
.
.
.
.
.
.
.
.
O A
(2)
B
(2)
C
(2)
O ...
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
O O ... A
(K)
B
(K)
C
(K)
O ...
.
.
.
O O ... O A
(K)
B
(K)
C
(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
th
row, j
th
column of a matrix M , then
the block matrices in Q can be deﬁned by the sys-
tem parameters λ
p
, λ
t
, µ
1
, µ
2
and α, as shown in the
Appendix.
Letting Q
∗
= A
(K)
+B
(K)
+C
(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
2
/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
49