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Tail asymptotics for a state-dependent bulk matching queueing system with impatient customers
Journal Pre-proof Tail asymptotics for a state dependent bulk matching queueing system with impatient customers
Qihui Bu, Yang Song, Liwei Liu
PII:
S0022-247X(19)31094-7
DOI:
https://doi.org/10.1016/j.jmaa.2019.123826
Reference:
YJMAA 123826
To appear in:
Journal of Mathematical Analysis and Applications
Received date:
19 August 2019
Please cite this article as: Q. Bu et al., Tail asymptotics for a state dependent bulk matching queueing system with impatient customers, J. Math. Anal. Appl. (2020), 123826, doi: https://doi.org/10.1016/j.jmaa.2019.123826.
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Tail asymptotics for a state dependent bulk matching queueing system with impatient customers Qihui Bu1 , Yang Song2 , Liwei Liu1,∗ 1: School of Science, Nanjing University of Science and Technology Nanjing 210094, Jiangsu, China 2: School of Science, Nanjing University of Aeronautics and Astronautics Nanjing 211106, Jiangsu, China ∗ Corresponding
Author: [email protected]
Abstract In this paper, we study a state dependent bulk matching queueing system with impatient customers, where customers and servers visit the system from both sides. Servers provide services in batch with a maximal size and take matching customers away instantly. For characterizing such a queueing system, the corresponding Markov process is constructed by the number of the complete batches of customers and the number of the remaining customers in the incomplete batch. By analyzing this system, we find it difficult to obtain the joint stationary distribution of the Markov process. Therefore, we pay attention to the tail asymptotics for the joint probabilities. Using the matrix analytic method and censoring technique, we obtain the one term and general expansions for the non-zero elements of the rate matrices, where the coefficients of expansions are presented in the closed form. Based on these expansion formulae, the exact tail asymptotic result for the joint stationary probabilities is derived. Keywords: Bulk matching queue; Censoring technique; Matrix-analytic method; Exact tail asymptotics; Series expansion.
1
Introduction
The so called matching queueing systems are characterized by the fact that customers and servers visit the system from both sides. When a server visits a system with waiting cus1
tomers, the server and its matching customers leave the system together. It is also applicable when a customer arrives at the system with waiting servers. Generally, such queueing systems arise from the modelling problems in transportation systems, computer networks, perishable inventory systems, organ transplantations and so on. Kendall [1] innovatively introduced a double-ended queueing system to investigate matching supply and demand problems in many stochastic service systems. Since then, many variants of the double-ended queueing system have been considered, which involve not only the system performances but also the optimal control problems and strategic behaviors. For example, Zenios [2] developed a matching queueing system with multiple classes of servers and customers, and finally obtained the closed-form asymptotic expressions for the stationary distribution of the waiting time. Mendoza et al. [3] studied the minimization of the expected total cost for a double-ended queue with thresholds on both sides. Wang et al. [4] paid attention to the strategic behaviors and social optimal strategies of incoming customers under three different information scenarios. In this paper, we assume that each server can provide a bulk service for customers with a fixed maximal batch size. Bulk service queueing systems have been extensively studied by many scholars. Nuets [5] pioneered the bulk service queueing system and proposed a general bulk service (GBS) rule. The general bulk service rule states that a server starts to provide a new service when at least ‘a’ customers present in the system, and the maximum service capacity is ‘b’. Chaudhry and Templeton [6] made further studies on the bulk service queueing systems, and gave a large number of theoretical results for readers to refer to. Other references listed but not limited for such queueing models are [7-10]. Meanwhile, customers’ impatience is considered in this paper since customers’ psychology is nonnegligible and important in real life. Taking impatience into account in queueing systems can trace back to 1957. Barrer [11, 12] discussed an M/M/1 queue and an M/M/c queue with constant impatience times, and obtained the steady-state queue length distributions for the both queueing systems, respectively. Baccelli et al. [13] considered a GI/GI/1 queueing system with impatient customers. They established the extensions of the classical GI/GI/1 formulae covering the stability condition, and the relations between actual and virtual waiting-time distribution functions. With the background of call centers, Zeltyn and Mandelbaum [14] studied an M/M/N + G queueing model under three asymptotic operational regimes, whose aim is to capture the tradeoff between the operational efficiency and
2
the service quality. For matching queueing systems, there seems to be little work done on the tail asymptotics of the stationary distribution, which is exactly the important topic of our research. Censoring technique is the key method used in this paper, which was usually adopted to characterize tail asymptotics of retrial queues in many literatures. One early work is Liu and Zhao [15] and the other one is Liu et al. [16]. The former considered a classical M/M/c retrial queue, and the later extended the retrial queue to the one with non-persistent customers. Respectively, Kim et al. [17], Kim and Kim [18] also studied the two models in [15] and [16]. Not only did they obtain the same decay functions, but also they refined the coefficients before the decay functions. Furthermore, Phung-Duc [19] applied a simple perturbation technique and matrix analytic method to derive the Taylor series expansions for the non-zero elements of rate matrices and the explicit expressions for all the coefficients of expansions. In this paper, we are devoted to study the tail asymptotics of a bulk matching queueing system by the matrix analytic method and censoring technique. The main contributions of this paper are two-folds: (1)We construct a bulk matching queueing system and restrict the servers to visit the system only when there are customers waiting in the system, which simplifies analyses and is reasonable in application. Furthermore, the bulk service model with impatient customers is an extension of the studies in the literature. (2)By the matrix analytic method and censoring technique, we obtain the asymptotic expansions for the nonzero entries of the rate matrices, which is with closed form expressions for coefficients in the expansions. Based on these expansions, the exact tail asymptotic result for the joint stationary probabilities is provided. The rest of this paper is organized as follows. In Section 2, a description for a bulk matching queueing system with impatient customers is carefully presented. Section 3 focuses on formulating the queueing system. The infinitesimal generator matrix is derived and a key property for the rate matrices is obtained by the censoring technique. In Section 4, we obtain series expansion formulae for all non-zero elements of the rate matrices and the exact decay functions for the joint stationary probabilities. Finally, concluding remarks are made in Section 5.
3
2
Model description and motivation
In this section, we introduce a state dependent bulk matching queueing system and provide an example of shuttle buses to emphasize the motivation and application of our work. We first describe a state dependent bulk matching queueing system with impatient customers, where customers and servers visit the system from both sides just as shown in Figure 1. Customers’ arrival process is assumed to be a Possion process with rate λ, while servers’ arrival process is statedependent and the arrival time intervals are exponential random varic0 , n = 0, , where c0 and c are positive constants, and n denotes ables with rate μn = cn, n = 0. the number of the complete batches of customers in the system. Suppose that services are provided in batch with a maximal size k. If there are Z customers in the system, then the Z . Servers visit the system only when number of the complete batches is given by n = k there are customers waiting and take matching customers away instantly, which means that only customers can join an empty system and it is not allowed for servers. In addition, customers’ impatient behaviour is considered in this work. Customers in the system can become impatient after an exponential time with rate α. In this paper, the impatient rate is assumed to be a constant no matter how many customers are waiting. For the case that every customer in the queue is allowed to be impatient independent of other customers, more discussions will be presented in the last section. The model proposed in our work has many applications in real life. Take the shuttle bus at the amusement park as an example. Each shuttle takes passengers as many as possible every time, and it has a fixed maximum capacity. The shuttles are dispatched according to the number of the waiting passengers at the station, which means the shuttle’s arrival depends on the passenger’s queue length. In general, passengers arrive at the station with no idea about the shuttle’s schedule, and they may lose patience when they are waiting.
3
Matrix formulation
In this section, we construct a continuous time Markov process to characterize the state dependent bulk matching queueing system. Applying the censoring technique and matrixanalytic method, an important lemma about the rate matrices is obtained, which is prepared for the proof of the tail asymptotics in the next section. 4
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Figure 1: Bulk matching queueing system with impatient customers Let N (t) be the number of the complete batches of customers in the system and X(t) be the number of the remaining customers in the incomplete batch at time t. Suppose that there Z(t) are Z(t) customers in the system at time t, then N (t) = and X(t) = Z(t) − k · N (t). k Obviously, (N (t), X(t)) is a continuous time Markov process with state space Ω = {(n, x) : n = 0, 1, 2, 3, . . . , and x = 0, 1, 2, . . . , k − 1}. It is easy to find that (N (t), X(t)) is a level dependent quasi-birth-and-death(QBD) process and the corresponding infinitesimal generator matrix is presented as follows, ⎛ ⎞ 0 A ⎜ ⎟ ⎜ C1 B 1 A ⎟ ⎜ ⎟ ⎜ ⎟ Q = ⎜ 0 C2 B 2 A ⎟, ⎜ ⎟ ⎜ 0 ⎟ C B A 3 3 ⎝ ⎠ .. .. .. .. . . . . where
⎛
−λ
⎞
λ
⎛
⎟ ⎜ ⎟ ⎜ ⎜ μ0 + α b0 λ ⎟ ⎜ ⎜ ⎟ ⎜ ⎜ α b0 λ 0 = ⎜ μ0 ⎟, A = ⎜ ⎜ ⎜ .. .. .. .. ⎟ ⎟ ⎜ ⎝ . . . . ⎠ ⎝ μ0 α b0 5
0 .. .
0 ... .. .
⎞
0 .. ⎟ ⎟ . ⎟ ⎟, 0 0 ... 0 ⎟ ⎠ λ 0 ... 0
(1)
for n = 0, 1, 2, . . .,
⎛
bn
⎜ ⎜ α ⎜ ⎜ Bn = ⎜ ⎜ ⎜ ⎝
⎞
λ bn λ .. .. .. . . . bn α
⎛
0 μ ⎟ ⎟ ⎜ n ⎟ ⎜ 0 μn ⎟ ⎜ ⎟ , Cn = ⎜ . ⎟ ⎜ .. ⎟ ⎝ λ ⎠ 0 0 bn
... 0 ... 0 .. . ...
α
⎞
⎟ 0 ⎟ ⎟ ⎟, ⎟ ⎠ μn
and the diagonal element bn of matrix Bn is equal to −(λ + μn + α). Under the condition λ < kμn for n = 0, 1, 2, . . ., the system is stable and we define the stationary joint probabilities of the number of the complete batches of customers and the number of the remaining customers in the incomplete batch, πn,x = lim P (N (t) = n, X(t) = x), n = 0, 1, 2, . . . ; x = 0, 1, . . . , k − 1. t→∞
Then, let πn = {πn,0 , πn,1 , . . . , πn,k−1 } and π = {π0 , π1 , π2 , . . .}. According to the matrix analytic method, we have the following results about the stationary distribution π, πQ = 0,
πe = 1,
πn = π0 R1 R2 . . . Rn , n = 1, 2, . . . , where Rn is called rate matrix. Adopting the censoring technique [15], we have ⎛ ⎞ A Bn ⎜ ⎟ ⎜ Cn+1 Bn+1 ⎟ A ⎜ ⎟ Qn = ⎜ ⎟, ⎜ ⎟ Cn+2 Bn+2 A ⎝ ⎠ ... ... ...
(2)
n represents the minimal non-negative inverse of −Qn and it is well n = (−Qn )−1 . Q and Q n (1, 1) as the (1, 1)-st block entry of Q n , then defined since that Qn is irreducible. Take Q for n = 1, 2, . . . , the rate matrix is determined by n (1, 1). Rn = AQ
(3)
Observing that matrix A has a special structure, which has only one non-zero element on the southwestern corner, thus, Rn can be further written as ⎞ ⎛ 0 0 ··· 0 ⎟ ⎜ ⎜ 0 0 ··· 0 ⎟ ⎟ ⎜ Rn = ⎜ . ⎟ , n = 1, 2, . . . . . . . .. .. .. ⎟ ⎜ .. ⎠ ⎝ rn,0 rn,1 · · · rn,k−1 6
(4)
Theoretically, we can calculate the stationary distribution of (N (t), X(t)) by the following theorem. Theorem 3.1. For the state dependent bulk matching queueing system with impatient customers , the stationary probability vector π is given by πn = π0,k−1 r1,k−1 r2,k−1 . . . rn−1,k−1 (rn,0 , rn,1 , . . . , rn,k−1 ),
n = 1, 2, . . . ,
(5)
and π0 can be uniquely determined by equation π0 0 + π1 C1 = 0 and the normalizing ∞ πn e = 1. equation n=0
However, there exist many difficulties during the solving process, such as how to obtain the inverse of infinite dimensional matrix −Qn . Therefore, rather than providing an algorithm for the solution of the probability vector, we turn to characterize the tail asymptotic behaviors of the stationary distribution, which will be carried out in the next section. Before ending this section, we give an important property of Rn in the following lemma, which will be used to prove the tail properties for the stationary distribution of (N (t), X(t)) in the next section. Lemma 3.1. For the state dependent bulk matching queueing system with impatient customers, we have
α 1+ μn
rn,0 + rn,1 + · · · + rn,k−1 =
λ . μn
(6)
¯ (t), X(t)) ¯ Proof. Consider a censored Markov chain (N with state space ¯ = {(¯ Ω n, x¯) : n ¯ = 0, 1, 2, 3, . . . , n − 1, and x¯ = 0, 1, 2, . . . , k − 1}. The corresponding generator matrix, denoted by Q≤(n−1) , is provided by ⎛ ⎞ 0 A ⎜ ⎟ ⎜ C1 B 1 A ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 C 2 B2 A ⎟ ≤(n−1) ⎟. Q =⎜ ... ... ... ⎜ .. ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ 0 ⎟ B A n−2 ⎝ ⎠ 0 Cn−1 Bn−1 + Rn Cn
7
(7)
Therefore, we have (Cn−1 + Bn−1 + Rn Cn )e = 0.
(8)
Since (Cn−1 + Bn−1 + A)e = 0, we have Rn Cn e = Ae. Expanding it yields that rn,0 μn + rn,1 μn + · · · + rn,k−1 μn + rn,0 α = λ,
(9)
which is equivalent to equation (6).
4
Tail analysis
In this section, we focus on the tail analysis of (N (t), X(t)), i.e. asymptotics for πn,x as n → ∞. The expansions with one term and M (M ∈ {2, 3, 4, . . .}) terms for rn,j are investigated. Based on these expansions, we derive the exact decay functions for the stationary joint probabilities of (N (t), X(t)). Define o(x) as the function of x such that limx→0 o(x)/x = 0 and O(x) as the function of x such that limx→0 O(x)/x = C, where C is a non-zero constant. By the matrix analytic method and (1), we have πn−1 A + πn Bn + πn+1 Cn+1 = 0,
(10)
A + Rn Bn + Rn Rn+1 Cn+1 = 0.
(11)
In matrix equation (11), we compare the both sides and get bn rn,0 + αrn,1 + μn+1 rn,k−1 rn+1,0 = −λ,
(12)
λrn,j−1 + bn rn,j + αrn,j+1 + μn+1 rn,k−1 rn+1,j = 0, j = 1, 2, . . . , k − 2, λrn,k−2 + bn rn,k−1 + μn+1 rn,k−1 rn+1,k−1 + αrn,k−1 rn+1,0 = 0.
(13) (14)
According to Lemma 3.1 and equations (12)–(14), the following lemmas are presented to study the tail asymptotic results for the stationary distribution of (N (t), X(t)). Lemma 4.1. For m = 1, 2, . . . , k − 1, we have
m+1
λ 1 1 , +o rn,m = c nm+1 nm+1 8
(15)
and lim (μn )m rn,j = 0,
(16)
n→∞
for j = m, m + 1, . . . , k − 1. Proof. First, according to Lemma 3.1, we know that lim rn,j = 0,
for j = 0, 1, 2, . . . , k − 1.
n→∞
(17)
Then, let n → ∞ in equation (14) and we have lim μn rn,k−1 = lim cnrn,k−1 = 0.
n→∞
n→∞
(18)
Recursively using equation (13) yields that lim μn rn,j = lim cnrn,j = 0,
n→∞
n→∞
for j = 1, 2, . . . , k − 2.
(19)
Substituting the above results into equation (12), we have lim μn rn,0 = λ and rn,0
λ1 = +o cn
1 . n
n→∞
(20)
Now, we have proved that equation (15) is true for rn,0 and (16) holds for m = 0, 1. Next, we will use mathematical induction to prove that rn,m also has the one term expansion like equation (15) and equation (16) is true for m = 2, 3, . . .. Assume that Lemma 4.1 is true for m = l − 1, l = 1, 2, . . . , k, which implies that
l
1 λ 1 rn,l−1 = +o , l c n nl
(21)
and lim (μn )l−1 rn,j = 0 for j = l − 1, l, . . . , k − 1. We will prove that is also true for m = l. n→∞
Multiplying equation (13) and (14) by (μn )l−1 and applying the results in the above assumption that lim (μn )l−1 rn,j = 0 for j = l − 1, l, . . . , k − 1, we can obtain n→∞
lim (μn )l rn,j = 0,
n→∞
for j = l, l + 1, . . . , k − 1.
(22)
Now, we have proved that equation (16) is true for m = 0, 1, 2, . . .. By equation (13) and (14), we have rn,l =
λrn,l−1 αrn,l+1 μn+1 rn,k−1 rn+1,l (λ + α)rn,l + + − , μn μn μn μn 9
(23)
for l = 1, 2, . . . , k − 2, and rn,k−1 =
λrn,k−2 αrn,k−1 rn+1,k−1 μn+1 rn,k−1 rn+1,k−1 (λ + α)rn,k−1 + + − . μn μn μn μn
(24)
Taking n → ∞ in equation (23) and analyzing the terms on the right side of the equation one by one, we have λ λrn,l−1 = μn μn
l+1 l λ 1 1 λ 1 1 = +o +o , c nl nl+1 c nl+1 nl+1
αrn,l+1 αrn,l+1 nl = =o μn μn n l
1
,
nl+1
μn+1 rn,k−1 rn+1,l μn+1 rn,k−1 rn+1,l nl = =o μn μn n l (λ + α)rn,l nl (λ + α)rn,l = =o μn μn n l
(25)
(26)
1 nl+1
1 nl+1
,
(27)
.
(28)
From equations (25)–(28), we verify that equation (15) is correct for m = 1, 2, . . . , k − 2. Finally, a similar analysis can be conducted for equation (24), and the correctness of (15) for m = k − 1 is proved. Actually, Lemma 4.1 can be further refined by using O(·) instead of o(·). The refined result is stated as the following lemma. Lemma 4.2. For m = 0, 1, . . . , k − 1,
m+1
λ 1 1 . +O rn,m = c nm+1 nm+2
(29)
Proof. Mathematical induction is employed again to prove this lemma. First, we affirm that this lemma holds for m = 0. Rewrite equation (6) as λ μn = − rn,j . α + μn μn + α j=1 k−1
rn,0
Using the results in Lemma 4.1, we conclude that
1 λ1 rn,0 = +O . cn n2 10
(30)
(31)
Then, assume that Lemma 4.2 holds for m = l − 1, l = 1, 2, . . . , k, i.e.
l
λ 1 1 . +O rn,l−1 = c nl nl+1 Applying the above assumptions and equations (23)-(24), we obtain
l+1
1 λ 1 rn,l = . +O c nl+1 nl+2
(32)
(33)
Thus, Lemma.4.2 is proved. The above two lemmas provide the one term expansions for the non-zero elements of Rn . Define γ1,m = (λ/c)m+1 , m = 0, 1, 2 . . . , k − 1. We improve Lemma 4.2 by expanding rn,m with M terms as follows, M ∈ {2, 3, 4, . . .}. Theorem 4.1. For M = 2, 3, . . . and m = 0, 1, 2, . . . , k − 1, we have rn,m =
M
γi,m (−1)
i+1
i=1
1 ni+m
+O
1
,
nM +m+1
(34)
where γM,0 = γM,m =
k−1 m=1 M −k
γM −m,m (−1)1−m , γi,k−1
M −i−k+1 s=1
i=1
(35)
(s + m − 1)u γs,m u!
λ (−1)−k + γM,m−1 c
α λ+α γM −2,m−1 + γM −1,m , m = 1, 2, . . . , k − 2, c c M −k M −i−k+1 (s + k − 2)u λ (−1)−k + γM,k−2 = γi,k−1 γs,k−1 u! c s=1 i=1 +
γM,k−1
+
M −k M −i−k λ+α α (s + k − 1)v γM −1,k−1 + , γi,k−1 γs,0 c c i=1 v! s=1
(36)
(37)
where u = M − i − k − s + 1 and v = M − i − k − s. Proof. We first prove the M −th term expansion γM,0 of rn,0 . Assume that we have known the expansion for rn,m with M − 1 terms (m = 0, 1, 2, . . . , k − 1), rn,m =
M −1 i=1
γi,m (−1)
i+1
1 ni+m
11
+O
1 nM +m
.
(38)
Substituting equation (38) into equation (30) and collecting the coefficients of λn−1 = − α + μn m=1 k−1
rn,0
M −1
γi,m (−1)i+1
i=1
1 ni+m
+ O(
1 nM +1
1 , we have nM
),
and (−1)
M +1
γM,0 = − γM,0 =
k−1
γM −m,m (−1)M −m+1 ,
(39)
m=1 k−1
γM −m,m (−1)−m+1 .
(40)
m=1
Furthermore, for j = 0, 1, 2, . . . , m − 1, assuming that the expansion for rn,j with M terms is known, we have rn,m−1 = rn,m+1 = rn,k−1 = rn+1,m = rn,m =
M
i+1
1
1
, M +m n i=1
M −1 1 1 i+1 , γi,m+1 (−1) +O i+m+1 M +m+1 n n i=1
M −1 1 1 i+1 , γi,k−1 (−1) +O ni+k−1 nM +k−1 i=1
M −1 1 1 i+1 , γi,m (−1) +O (1 + n)i+m nM +m i=1
M −1 1 i+1 1 . γi,m (−1) +O ni+m nM +m i=1 γi,m−1 (−1)
ni+m−1
+O
(41) (42) (43) (44) (45)
Multiplying (1 + n) on the both two sides of equation (44) and applying the fact that
1 1+ n
−b =
∞ (b)s s=0
s!
(−1)s
1 , ns
(46)
we obtain (1 + n)rn+1,m =
M −1
=
i=1
1
1 1+ n
−(i+m−1)
1
+O ni+m−1 nM +m−1 M −i−1
(i + m − 1)j 1 1 1 i+1 j . (−1) j +O γi,m (−1) M +m−1 ni+m−1 j! n n j=0
i=1 M −1
γi,m (−1)
i+1
12
Substituting the above equations into equations (23)-(24) and comparing the coefficient of 1 , we come to the conclusions. M n +m Based on the above asymptotic results of rn,m , we give the tail asymptotic analysis for the stationary probabilities πn,x . Theorem 4.2. For the state dependent bulk matching queueing system with impatient customers, the exact decay function for the stationary probability πn,x is provided by n hx (n) = γ1,k−1
1 −φ−(x+1)+k n , n!
n = 0, 1, . . . ,
(47)
for x = 0, 1, . . . , k − 1. That is, there exit two positive constants independent of n, C1 and C2 , satisfying that n C1 γ1,k−1
where φ = (k − 1)
1 −φ−(x+1)+k 1 −φ−(x+1)+k n n n < πx,n < C2 γ1,k−1 , n! n!
(48)
λ+α λ + . c c
Proof. According to Theorem 4.1, we have the expansion with three terms for rn,k−1 as follows, rn,k−1
1 1 1 1 , = γ1,k−1 k 1 − φ + ϕ 2 + O n n n nk+3
φ2 λ+α λ + ,ϕ= + κ and c c 2
2 k−1 λ+α (λ)2 λ(λ + α) α κ= − 2 + (k − 1) + (k − 2) > 0. 2 2 c 2c c λ
(49)
where φ = (k − 1)
(50)
Since φ2 − 4ϕ < 0, using Theorem 3.2 in Liu and Zhao [16], we have n 0 < C1 γ1,k−1
1 −φ 1 −φ n n < r1,k−1 r2,k−1 , . . . , rn,k−1 < C2 γ1,k−1 n , n! n!
(51)
where C1 and C2 are positive constant independent of n. Then, according to Lemma.4.2 and rn,x πx,n = , we have πk−1,n rn,k−1 n C1 γ1,k−1
1 −φ−(x+1)+k 1 −φ−(x+1)+k n n n < πx,n < C2 γ1,k−1 . n! n!
Thus, we conclude the desired results in Theorem 4.2. 13
(52)
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([DFWGHFD\IXQFWLRQ K [ Q
1XPEHURIFRPSOHWHEDWFKHV Q
Figure 2: The exact decay function versus n for different value of x Example 4.1. Consider the bulk matching queueing system described in this paper with a maximal batch size k = 5. Then, the exact decay function degrades into
5λ + 4α
5n −x+4 1 − λ c hx (n) = n , x = 0, 1, 2, 3, 4. c n!
(53)
Let λ = 0.5, μn = 0.4n, α = 0.3. As the number of complete batches in the system increases from 0 to 100, the trend of the change for hx (n) is shown in Figure 2. Observe that the curves are asymptotically linear in the logarithmic scale for the vertical axis. According to the bounds shown in equation (52), for a given N0 , when n > N0 , the stationary probability πn,x can approximated by the decay function perfectly.
5
Concluding remarks
In this paper, we investigate the tail asymptotics for a state dependent bulk matching queue with impatient customers by the matrix analytic method and censoring technique. An 14
important property and general asymptotic expansions for the non-zero entries of the rate matrices are obtained. According to these expansions, we further derive the exact decay functions for the stationary probabilities of (N (t), X(t)). In addition, we assume that every customer may get impatient in this bulk matching queueing system , and the impatient time for each customer is an exponential variable with 1 mean . Then, the impatient rate at state (n, x) is αnx = (nk + x)α. Following the same α idea and method, we make some minor revisions in Lemma 3.1 and Lemma 4.1, respectively, λ αn+1,0 rn,0 + rn,1 + · · · + rn,k−1 = , 1+ μn μn m+1
λ 1 1 , +o rn,m = c + kα nm+1 nm+1
and lim nm rn,j = 0 for j = m, m + 1, . . .. It is difficult and challenging to derive the n→∞
general expansions for rn,m , even the expansions with two terms. Without the asymptotic expansions, the tail behaviours for the stationary joint probabilities are hard to be conducted. Even so, the tail asymptotic behaviour of a bulk matching queueing system with such linear impatient rate is an interesting problem and we will try to study it in the future work.
Acknowledgement This work was supported by the National Natural Science Foundation of China (No.61773014), the Postgraduate Research and Practice Innovation Program of Jiangsu Province (No.KYCX19 0250), the Natural Science Foundation of Jiangsu Province (No.BK20160788) and the Fundamental Research Funds for the Central Universities (No.NS2017050).
References [1] Kendall, D. G. (1951). Some problems in the theory of queues. Journal of the Royal Statistical Society: Series B (Methodological), 13(2), 151–173. [2] Zenios, S. A. (1999). Modeling the transplant waiting list: A queueing model with reneging. Queueing Systems, 31(3-4), 239–251. [3] Mendoza, G., Sedaghat, M. and Yoon, K. P. (2009). Queuing models to balance systems with excess supply. International Business & Economics Research Journal, 8(1), 91–104.
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[4] 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. [5] Neuts, M. F. (1967). A general class of bulk queues with Poisson input. The Annals of Mathematical Statistics, 38(3), 759–770. [6] Chaudhry, M. L. and Templeton, J. G. (1983). A first course in bulk queues (pp. 2001–2004). New York: Wiley. [7] Zhao, Y. Q. and Campbell, L. L. (1996). Equilibrium probability calculations for a discretetime bulk queue model. Queueing Systems, 22(1–2), 189–198. [8] Bar-Lev, S. K., Parlar, M., Perry, D., Stadje, W. and Van der Duyn Schouten, F. A. (2007). Applications of bulk queues to group testing models with incomplete identification. European Journal of Operational Research, 183(1), 226–237. [9] Claeys, D., Walraevens, J., Laevens, K. and Bruneel, H. (2011). Analysis of threshold-based batch-service queueing systems with batch arrivals and general service times. Performance Evaluation, 68(6), 528–549. [10] Banerjee, A., Gupta, U. C. and Chakravarthy, S. R. (2015). Analysis of a finite-buffer bulkservice queue under Markovian arrival process with batch-size-dependent service. Computers and Operations Research, 60, 138–149. [11] Barrer, D. Y. (1957a). Queueing with impatient customers and indifferent clerks. Operations Research, 5, 644—649. [12] Barrer, D. Y. (1957b). Queueing with impatient customers and ordered service. Operations Research, 5, 650–656. [13] Baccelli, F., Boyer, P. and Hebuterne, G. (1984). Single-server queues with impatient customers. Advances in Applied Probability, 16(4), 887–905. [14] Zeltyn, S. and Mandelbaum, A. (2005). Call centers with impatient customers: many-server asymptotics of the M/M/n + G queue. Queueing Systems, 51(3-4), 361–402. [15] Liu, B., Zhao, Y.Q. (2010). Analyzing retrial queues by censoring. Queueing System, 64(3), 203–225 . [16] Liu, B., Wang, X. and Zhao, Y.Q. (2012). Tail asymptotics for M/M/c retrial queues with non-persistent customers, Opereational Research, 12, 173—188. [17] Kim, J., Kim, J. and Kim, B. (2012). Tail asymptotics of the queue size distribution in the M/M/m retrial queue. Journal of Computational and Applied Mathematics, 236(14), 3445– 3460. [18] Kim, B., Kim, J. (2012). Exact tail asymptotics for the M/M/m retrial queue with nonpersistent customers. Operations Research Letters, 40(6), 537–540. [19] Phung-Duc, T. (2015). Asymptotic analysis for Markovian queues with two types of nonpersistent retrial customers, Applied Mathematics and Computation, 265, 768–784.
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Qihui Bu, Yang Song, Liwei Liu
PII:
S0022-247X(19)31094-7
DOI:
https://doi.org/10.1016/j.jmaa.2019.123826
Reference:
YJMAA 123826
To appear in:
Journal of Mathematical Analysis and Applications
Received date:
19 August 2019
Please cite this article as: Q. Bu et al., Tail asymptotics for a state dependent bulk matching queueing system with impatient customers, J. Math. Anal. Appl. (2020), 123826, doi: https://doi.org/10.1016/j.jmaa.2019.123826.
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Tail asymptotics for a state dependent bulk matching queueing system with impatient customers Qihui Bu1 , Yang Song2 , Liwei Liu1,∗ 1: School of Science, Nanjing University of Science and Technology Nanjing 210094, Jiangsu, China 2: School of Science, Nanjing University of Aeronautics and Astronautics Nanjing 211106, Jiangsu, China ∗ Corresponding
Author: [email protected]
Abstract In this paper, we study a state dependent bulk matching queueing system with impatient customers, where customers and servers visit the system from both sides. Servers provide services in batch with a maximal size and take matching customers away instantly. For characterizing such a queueing system, the corresponding Markov process is constructed by the number of the complete batches of customers and the number of the remaining customers in the incomplete batch. By analyzing this system, we find it difficult to obtain the joint stationary distribution of the Markov process. Therefore, we pay attention to the tail asymptotics for the joint probabilities. Using the matrix analytic method and censoring technique, we obtain the one term and general expansions for the non-zero elements of the rate matrices, where the coefficients of expansions are presented in the closed form. Based on these expansion formulae, the exact tail asymptotic result for the joint stationary probabilities is derived. Keywords: Bulk matching queue; Censoring technique; Matrix-analytic method; Exact tail asymptotics; Series expansion.
1
Introduction
The so called matching queueing systems are characterized by the fact that customers and servers visit the system from both sides. When a server visits a system with waiting cus1
tomers, the server and its matching customers leave the system together. It is also applicable when a customer arrives at the system with waiting servers. Generally, such queueing systems arise from the modelling problems in transportation systems, computer networks, perishable inventory systems, organ transplantations and so on. Kendall [1] innovatively introduced a double-ended queueing system to investigate matching supply and demand problems in many stochastic service systems. Since then, many variants of the double-ended queueing system have been considered, which involve not only the system performances but also the optimal control problems and strategic behaviors. For example, Zenios [2] developed a matching queueing system with multiple classes of servers and customers, and finally obtained the closed-form asymptotic expressions for the stationary distribution of the waiting time. Mendoza et al. [3] studied the minimization of the expected total cost for a double-ended queue with thresholds on both sides. Wang et al. [4] paid attention to the strategic behaviors and social optimal strategies of incoming customers under three different information scenarios. In this paper, we assume that each server can provide a bulk service for customers with a fixed maximal batch size. Bulk service queueing systems have been extensively studied by many scholars. Nuets [5] pioneered the bulk service queueing system and proposed a general bulk service (GBS) rule. The general bulk service rule states that a server starts to provide a new service when at least ‘a’ customers present in the system, and the maximum service capacity is ‘b’. Chaudhry and Templeton [6] made further studies on the bulk service queueing systems, and gave a large number of theoretical results for readers to refer to. Other references listed but not limited for such queueing models are [7-10]. Meanwhile, customers’ impatience is considered in this paper since customers’ psychology is nonnegligible and important in real life. Taking impatience into account in queueing systems can trace back to 1957. Barrer [11, 12] discussed an M/M/1 queue and an M/M/c queue with constant impatience times, and obtained the steady-state queue length distributions for the both queueing systems, respectively. Baccelli et al. [13] considered a GI/GI/1 queueing system with impatient customers. They established the extensions of the classical GI/GI/1 formulae covering the stability condition, and the relations between actual and virtual waiting-time distribution functions. With the background of call centers, Zeltyn and Mandelbaum [14] studied an M/M/N + G queueing model under three asymptotic operational regimes, whose aim is to capture the tradeoff between the operational efficiency and
2
the service quality. For matching queueing systems, there seems to be little work done on the tail asymptotics of the stationary distribution, which is exactly the important topic of our research. Censoring technique is the key method used in this paper, which was usually adopted to characterize tail asymptotics of retrial queues in many literatures. One early work is Liu and Zhao [15] and the other one is Liu et al. [16]. The former considered a classical M/M/c retrial queue, and the later extended the retrial queue to the one with non-persistent customers. Respectively, Kim et al. [17], Kim and Kim [18] also studied the two models in [15] and [16]. Not only did they obtain the same decay functions, but also they refined the coefficients before the decay functions. Furthermore, Phung-Duc [19] applied a simple perturbation technique and matrix analytic method to derive the Taylor series expansions for the non-zero elements of rate matrices and the explicit expressions for all the coefficients of expansions. In this paper, we are devoted to study the tail asymptotics of a bulk matching queueing system by the matrix analytic method and censoring technique. The main contributions of this paper are two-folds: (1)We construct a bulk matching queueing system and restrict the servers to visit the system only when there are customers waiting in the system, which simplifies analyses and is reasonable in application. Furthermore, the bulk service model with impatient customers is an extension of the studies in the literature. (2)By the matrix analytic method and censoring technique, we obtain the asymptotic expansions for the nonzero entries of the rate matrices, which is with closed form expressions for coefficients in the expansions. Based on these expansions, the exact tail asymptotic result for the joint stationary probabilities is provided. The rest of this paper is organized as follows. In Section 2, a description for a bulk matching queueing system with impatient customers is carefully presented. Section 3 focuses on formulating the queueing system. The infinitesimal generator matrix is derived and a key property for the rate matrices is obtained by the censoring technique. In Section 4, we obtain series expansion formulae for all non-zero elements of the rate matrices and the exact decay functions for the joint stationary probabilities. Finally, concluding remarks are made in Section 5.
3
2
Model description and motivation
In this section, we introduce a state dependent bulk matching queueing system and provide an example of shuttle buses to emphasize the motivation and application of our work. We first describe a state dependent bulk matching queueing system with impatient customers, where customers and servers visit the system from both sides just as shown in Figure 1. Customers’ arrival process is assumed to be a Possion process with rate λ, while servers’ arrival process is statedependent and the arrival time intervals are exponential random varic0 , n = 0, , where c0 and c are positive constants, and n denotes ables with rate μn = cn, n = 0. the number of the complete batches of customers in the system. Suppose that services are provided in batch with a maximal size k. If there are Z customers in the system, then the Z . Servers visit the system only when number of the complete batches is given by n = k there are customers waiting and take matching customers away instantly, which means that only customers can join an empty system and it is not allowed for servers. In addition, customers’ impatient behaviour is considered in this work. Customers in the system can become impatient after an exponential time with rate α. In this paper, the impatient rate is assumed to be a constant no matter how many customers are waiting. For the case that every customer in the queue is allowed to be impatient independent of other customers, more discussions will be presented in the last section. The model proposed in our work has many applications in real life. Take the shuttle bus at the amusement park as an example. Each shuttle takes passengers as many as possible every time, and it has a fixed maximum capacity. The shuttles are dispatched according to the number of the waiting passengers at the station, which means the shuttle’s arrival depends on the passenger’s queue length. In general, passengers arrive at the station with no idea about the shuttle’s schedule, and they may lose patience when they are waiting.
3
Matrix formulation
In this section, we construct a continuous time Markov process to characterize the state dependent bulk matching queueing system. Applying the censoring technique and matrixanalytic method, an important lemma about the rate matrices is obtained, which is prepared for the proof of the tail asymptotics in the next section. 4
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Figure 1: Bulk matching queueing system with impatient customers Let N (t) be the number of the complete batches of customers in the system and X(t) be the number of the remaining customers in the incomplete batch at time t. Suppose that there Z(t) are Z(t) customers in the system at time t, then N (t) = and X(t) = Z(t) − k · N (t). k Obviously, (N (t), X(t)) is a continuous time Markov process with state space Ω = {(n, x) : n = 0, 1, 2, 3, . . . , and x = 0, 1, 2, . . . , k − 1}. It is easy to find that (N (t), X(t)) is a level dependent quasi-birth-and-death(QBD) process and the corresponding infinitesimal generator matrix is presented as follows, ⎛ ⎞ 0 A ⎜ ⎟ ⎜ C1 B 1 A ⎟ ⎜ ⎟ ⎜ ⎟ Q = ⎜ 0 C2 B 2 A ⎟, ⎜ ⎟ ⎜ 0 ⎟ C B A 3 3 ⎝ ⎠ .. .. .. .. . . . . where
⎛
−λ
⎞
λ
⎛
⎟ ⎜ ⎟ ⎜ ⎜ μ0 + α b0 λ ⎟ ⎜ ⎜ ⎟ ⎜ ⎜ α b0 λ 0 = ⎜ μ0 ⎟, A = ⎜ ⎜ ⎜ .. .. .. .. ⎟ ⎟ ⎜ ⎝ . . . . ⎠ ⎝ μ0 α b0 5
0 .. .
0 ... .. .
⎞
0 .. ⎟ ⎟ . ⎟ ⎟, 0 0 ... 0 ⎟ ⎠ λ 0 ... 0
(1)
for n = 0, 1, 2, . . .,
⎛
bn
⎜ ⎜ α ⎜ ⎜ Bn = ⎜ ⎜ ⎜ ⎝
⎞
λ bn λ .. .. .. . . . bn α
⎛
0 μ ⎟ ⎟ ⎜ n ⎟ ⎜ 0 μn ⎟ ⎜ ⎟ , Cn = ⎜ . ⎟ ⎜ .. ⎟ ⎝ λ ⎠ 0 0 bn
... 0 ... 0 .. . ...
α
⎞
⎟ 0 ⎟ ⎟ ⎟, ⎟ ⎠ μn
and the diagonal element bn of matrix Bn is equal to −(λ + μn + α). Under the condition λ < kμn for n = 0, 1, 2, . . ., the system is stable and we define the stationary joint probabilities of the number of the complete batches of customers and the number of the remaining customers in the incomplete batch, πn,x = lim P (N (t) = n, X(t) = x), n = 0, 1, 2, . . . ; x = 0, 1, . . . , k − 1. t→∞
Then, let πn = {πn,0 , πn,1 , . . . , πn,k−1 } and π = {π0 , π1 , π2 , . . .}. According to the matrix analytic method, we have the following results about the stationary distribution π, πQ = 0,
πe = 1,
πn = π0 R1 R2 . . . Rn , n = 1, 2, . . . , where Rn is called rate matrix. Adopting the censoring technique [15], we have ⎛ ⎞ A Bn ⎜ ⎟ ⎜ Cn+1 Bn+1 ⎟ A ⎜ ⎟ Qn = ⎜ ⎟, ⎜ ⎟ Cn+2 Bn+2 A ⎝ ⎠ ... ... ...
(2)
n represents the minimal non-negative inverse of −Qn and it is well n = (−Qn )−1 . Q and Q n (1, 1) as the (1, 1)-st block entry of Q n , then defined since that Qn is irreducible. Take Q for n = 1, 2, . . . , the rate matrix is determined by n (1, 1). Rn = AQ
(3)
Observing that matrix A has a special structure, which has only one non-zero element on the southwestern corner, thus, Rn can be further written as ⎞ ⎛ 0 0 ··· 0 ⎟ ⎜ ⎜ 0 0 ··· 0 ⎟ ⎟ ⎜ Rn = ⎜ . ⎟ , n = 1, 2, . . . . . . . .. .. .. ⎟ ⎜ .. ⎠ ⎝ rn,0 rn,1 · · · rn,k−1 6
(4)
Theoretically, we can calculate the stationary distribution of (N (t), X(t)) by the following theorem. Theorem 3.1. For the state dependent bulk matching queueing system with impatient customers , the stationary probability vector π is given by πn = π0,k−1 r1,k−1 r2,k−1 . . . rn−1,k−1 (rn,0 , rn,1 , . . . , rn,k−1 ),
n = 1, 2, . . . ,
(5)
and π0 can be uniquely determined by equation π0 0 + π1 C1 = 0 and the normalizing ∞ πn e = 1. equation n=0
However, there exist many difficulties during the solving process, such as how to obtain the inverse of infinite dimensional matrix −Qn . Therefore, rather than providing an algorithm for the solution of the probability vector, we turn to characterize the tail asymptotic behaviors of the stationary distribution, which will be carried out in the next section. Before ending this section, we give an important property of Rn in the following lemma, which will be used to prove the tail properties for the stationary distribution of (N (t), X(t)) in the next section. Lemma 3.1. For the state dependent bulk matching queueing system with impatient customers, we have
α 1+ μn
rn,0 + rn,1 + · · · + rn,k−1 =
λ . μn
(6)
¯ (t), X(t)) ¯ Proof. Consider a censored Markov chain (N with state space ¯ = {(¯ Ω n, x¯) : n ¯ = 0, 1, 2, 3, . . . , n − 1, and x¯ = 0, 1, 2, . . . , k − 1}. The corresponding generator matrix, denoted by Q≤(n−1) , is provided by ⎛ ⎞ 0 A ⎜ ⎟ ⎜ C1 B 1 A ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 C 2 B2 A ⎟ ≤(n−1) ⎟. Q =⎜ ... ... ... ⎜ .. ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ 0 ⎟ B A n−2 ⎝ ⎠ 0 Cn−1 Bn−1 + Rn Cn
7
(7)
Therefore, we have (Cn−1 + Bn−1 + Rn Cn )e = 0.
(8)
Since (Cn−1 + Bn−1 + A)e = 0, we have Rn Cn e = Ae. Expanding it yields that rn,0 μn + rn,1 μn + · · · + rn,k−1 μn + rn,0 α = λ,
(9)
which is equivalent to equation (6).
4
Tail analysis
In this section, we focus on the tail analysis of (N (t), X(t)), i.e. asymptotics for πn,x as n → ∞. The expansions with one term and M (M ∈ {2, 3, 4, . . .}) terms for rn,j are investigated. Based on these expansions, we derive the exact decay functions for the stationary joint probabilities of (N (t), X(t)). Define o(x) as the function of x such that limx→0 o(x)/x = 0 and O(x) as the function of x such that limx→0 O(x)/x = C, where C is a non-zero constant. By the matrix analytic method and (1), we have πn−1 A + πn Bn + πn+1 Cn+1 = 0,
(10)
A + Rn Bn + Rn Rn+1 Cn+1 = 0.
(11)
In matrix equation (11), we compare the both sides and get bn rn,0 + αrn,1 + μn+1 rn,k−1 rn+1,0 = −λ,
(12)
λrn,j−1 + bn rn,j + αrn,j+1 + μn+1 rn,k−1 rn+1,j = 0, j = 1, 2, . . . , k − 2, λrn,k−2 + bn rn,k−1 + μn+1 rn,k−1 rn+1,k−1 + αrn,k−1 rn+1,0 = 0.
(13) (14)
According to Lemma 3.1 and equations (12)–(14), the following lemmas are presented to study the tail asymptotic results for the stationary distribution of (N (t), X(t)). Lemma 4.1. For m = 1, 2, . . . , k − 1, we have
m+1
λ 1 1 , +o rn,m = c nm+1 nm+1 8
(15)
and lim (μn )m rn,j = 0,
(16)
n→∞
for j = m, m + 1, . . . , k − 1. Proof. First, according to Lemma 3.1, we know that lim rn,j = 0,
for j = 0, 1, 2, . . . , k − 1.
n→∞
(17)
Then, let n → ∞ in equation (14) and we have lim μn rn,k−1 = lim cnrn,k−1 = 0.
n→∞
n→∞
(18)
Recursively using equation (13) yields that lim μn rn,j = lim cnrn,j = 0,
n→∞
n→∞
for j = 1, 2, . . . , k − 2.
(19)
Substituting the above results into equation (12), we have lim μn rn,0 = λ and rn,0
λ1 = +o cn
1 . n
n→∞
(20)
Now, we have proved that equation (15) is true for rn,0 and (16) holds for m = 0, 1. Next, we will use mathematical induction to prove that rn,m also has the one term expansion like equation (15) and equation (16) is true for m = 2, 3, . . .. Assume that Lemma 4.1 is true for m = l − 1, l = 1, 2, . . . , k, which implies that
l
1 λ 1 rn,l−1 = +o , l c n nl
(21)
and lim (μn )l−1 rn,j = 0 for j = l − 1, l, . . . , k − 1. We will prove that is also true for m = l. n→∞
Multiplying equation (13) and (14) by (μn )l−1 and applying the results in the above assumption that lim (μn )l−1 rn,j = 0 for j = l − 1, l, . . . , k − 1, we can obtain n→∞
lim (μn )l rn,j = 0,
n→∞
for j = l, l + 1, . . . , k − 1.
(22)
Now, we have proved that equation (16) is true for m = 0, 1, 2, . . .. By equation (13) and (14), we have rn,l =
λrn,l−1 αrn,l+1 μn+1 rn,k−1 rn+1,l (λ + α)rn,l + + − , μn μn μn μn 9
(23)
for l = 1, 2, . . . , k − 2, and rn,k−1 =
λrn,k−2 αrn,k−1 rn+1,k−1 μn+1 rn,k−1 rn+1,k−1 (λ + α)rn,k−1 + + − . μn μn μn μn
(24)
Taking n → ∞ in equation (23) and analyzing the terms on the right side of the equation one by one, we have λ λrn,l−1 = μn μn
l+1 l λ 1 1 λ 1 1 = +o +o , c nl nl+1 c nl+1 nl+1
αrn,l+1 αrn,l+1 nl = =o μn μn n l
1
,
nl+1
μn+1 rn,k−1 rn+1,l μn+1 rn,k−1 rn+1,l nl = =o μn μn n l (λ + α)rn,l nl (λ + α)rn,l = =o μn μn n l
(25)
(26)
1 nl+1
1 nl+1
,
(27)
.
(28)
From equations (25)–(28), we verify that equation (15) is correct for m = 1, 2, . . . , k − 2. Finally, a similar analysis can be conducted for equation (24), and the correctness of (15) for m = k − 1 is proved. Actually, Lemma 4.1 can be further refined by using O(·) instead of o(·). The refined result is stated as the following lemma. Lemma 4.2. For m = 0, 1, . . . , k − 1,
m+1
λ 1 1 . +O rn,m = c nm+1 nm+2
(29)
Proof. Mathematical induction is employed again to prove this lemma. First, we affirm that this lemma holds for m = 0. Rewrite equation (6) as λ μn = − rn,j . α + μn μn + α j=1 k−1
rn,0
Using the results in Lemma 4.1, we conclude that
1 λ1 rn,0 = +O . cn n2 10
(30)
(31)
Then, assume that Lemma 4.2 holds for m = l − 1, l = 1, 2, . . . , k, i.e.
l
λ 1 1 . +O rn,l−1 = c nl nl+1 Applying the above assumptions and equations (23)-(24), we obtain
l+1
1 λ 1 rn,l = . +O c nl+1 nl+2
(32)
(33)
Thus, Lemma.4.2 is proved. The above two lemmas provide the one term expansions for the non-zero elements of Rn . Define γ1,m = (λ/c)m+1 , m = 0, 1, 2 . . . , k − 1. We improve Lemma 4.2 by expanding rn,m with M terms as follows, M ∈ {2, 3, 4, . . .}. Theorem 4.1. For M = 2, 3, . . . and m = 0, 1, 2, . . . , k − 1, we have rn,m =
M
γi,m (−1)
i+1
i=1
1 ni+m
+O
1
,
nM +m+1
(34)
where γM,0 = γM,m =
k−1 m=1 M −k
γM −m,m (−1)1−m , γi,k−1
M −i−k+1 s=1
i=1
(35)
(s + m − 1)u γs,m u!
λ (−1)−k + γM,m−1 c
α λ+α γM −2,m−1 + γM −1,m , m = 1, 2, . . . , k − 2, c c M −k M −i−k+1 (s + k − 2)u λ (−1)−k + γM,k−2 = γi,k−1 γs,k−1 u! c s=1 i=1 +
γM,k−1
+
M −k M −i−k λ+α α (s + k − 1)v γM −1,k−1 + , γi,k−1 γs,0 c c i=1 v! s=1
(36)
(37)
where u = M − i − k − s + 1 and v = M − i − k − s. Proof. We first prove the M −th term expansion γM,0 of rn,0 . Assume that we have known the expansion for rn,m with M − 1 terms (m = 0, 1, 2, . . . , k − 1), rn,m =
M −1 i=1
γi,m (−1)
i+1
1 ni+m
11
+O
1 nM +m
.
(38)
Substituting equation (38) into equation (30) and collecting the coefficients of λn−1 = − α + μn m=1 k−1
rn,0
M −1
γi,m (−1)i+1
i=1
1 ni+m
+ O(
1 nM +1
1 , we have nM
),
and (−1)
M +1
γM,0 = − γM,0 =
k−1
γM −m,m (−1)M −m+1 ,
(39)
m=1 k−1
γM −m,m (−1)−m+1 .
(40)
m=1
Furthermore, for j = 0, 1, 2, . . . , m − 1, assuming that the expansion for rn,j with M terms is known, we have rn,m−1 = rn,m+1 = rn,k−1 = rn+1,m = rn,m =
M
i+1
1
1
, M +m n i=1
M −1 1 1 i+1 , γi,m+1 (−1) +O i+m+1 M +m+1 n n i=1
M −1 1 1 i+1 , γi,k−1 (−1) +O ni+k−1 nM +k−1 i=1
M −1 1 1 i+1 , γi,m (−1) +O (1 + n)i+m nM +m i=1
M −1 1 i+1 1 . γi,m (−1) +O ni+m nM +m i=1 γi,m−1 (−1)
ni+m−1
+O
(41) (42) (43) (44) (45)
Multiplying (1 + n) on the both two sides of equation (44) and applying the fact that
1 1+ n
−b =
∞ (b)s s=0
s!
(−1)s
1 , ns
(46)
we obtain (1 + n)rn+1,m =
M −1
=
i=1
1
1 1+ n
−(i+m−1)
1
+O ni+m−1 nM +m−1 M −i−1
(i + m − 1)j 1 1 1 i+1 j . (−1) j +O γi,m (−1) M +m−1 ni+m−1 j! n n j=0
i=1 M −1
γi,m (−1)
i+1
12
Substituting the above equations into equations (23)-(24) and comparing the coefficient of 1 , we come to the conclusions. M n +m Based on the above asymptotic results of rn,m , we give the tail asymptotic analysis for the stationary probabilities πn,x . Theorem 4.2. For the state dependent bulk matching queueing system with impatient customers, the exact decay function for the stationary probability πn,x is provided by n hx (n) = γ1,k−1
1 −φ−(x+1)+k n , n!
n = 0, 1, . . . ,
(47)
for x = 0, 1, . . . , k − 1. That is, there exit two positive constants independent of n, C1 and C2 , satisfying that n C1 γ1,k−1
where φ = (k − 1)
1 −φ−(x+1)+k 1 −φ−(x+1)+k n n n < πx,n < C2 γ1,k−1 , n! n!
(48)
λ+α λ + . c c
Proof. According to Theorem 4.1, we have the expansion with three terms for rn,k−1 as follows, rn,k−1
1 1 1 1 , = γ1,k−1 k 1 − φ + ϕ 2 + O n n n nk+3
φ2 λ+α λ + ,ϕ= + κ and c c 2
2 k−1 λ+α (λ)2 λ(λ + α) α κ= − 2 + (k − 1) + (k − 2) > 0. 2 2 c 2c c λ
(49)
where φ = (k − 1)
(50)
Since φ2 − 4ϕ < 0, using Theorem 3.2 in Liu and Zhao [16], we have n 0 < C1 γ1,k−1
1 −φ 1 −φ n n < r1,k−1 r2,k−1 , . . . , rn,k−1 < C2 γ1,k−1 n , n! n!
(51)
where C1 and C2 are positive constant independent of n. Then, according to Lemma.4.2 and rn,x πx,n = , we have πk−1,n rn,k−1 n C1 γ1,k−1
1 −φ−(x+1)+k 1 −φ−(x+1)+k n n n < πx,n < C2 γ1,k−1 . n! n!
Thus, we conclude the desired results in Theorem 4.2. 13
(52)
[ [ [
([DFWGHFD\IXQFWLRQ K [ Q
1XPEHURIFRPSOHWHEDWFKHV Q
Figure 2: The exact decay function versus n for different value of x Example 4.1. Consider the bulk matching queueing system described in this paper with a maximal batch size k = 5. Then, the exact decay function degrades into
5λ + 4α
5n −x+4 1 − λ c hx (n) = n , x = 0, 1, 2, 3, 4. c n!
(53)
Let λ = 0.5, μn = 0.4n, α = 0.3. As the number of complete batches in the system increases from 0 to 100, the trend of the change for hx (n) is shown in Figure 2. Observe that the curves are asymptotically linear in the logarithmic scale for the vertical axis. According to the bounds shown in equation (52), for a given N0 , when n > N0 , the stationary probability πn,x can approximated by the decay function perfectly.
5
Concluding remarks
In this paper, we investigate the tail asymptotics for a state dependent bulk matching queue with impatient customers by the matrix analytic method and censoring technique. An 14
important property and general asymptotic expansions for the non-zero entries of the rate matrices are obtained. According to these expansions, we further derive the exact decay functions for the stationary probabilities of (N (t), X(t)). In addition, we assume that every customer may get impatient in this bulk matching queueing system , and the impatient time for each customer is an exponential variable with 1 mean . Then, the impatient rate at state (n, x) is αnx = (nk + x)α. Following the same α idea and method, we make some minor revisions in Lemma 3.1 and Lemma 4.1, respectively, λ αn+1,0 rn,0 + rn,1 + · · · + rn,k−1 = , 1+ μn μn m+1
λ 1 1 , +o rn,m = c + kα nm+1 nm+1
and lim nm rn,j = 0 for j = m, m + 1, . . .. It is difficult and challenging to derive the n→∞
general expansions for rn,m , even the expansions with two terms. Without the asymptotic expansions, the tail behaviours for the stationary joint probabilities are hard to be conducted. Even so, the tail asymptotic behaviour of a bulk matching queueing system with such linear impatient rate is an interesting problem and we will try to study it in the future work.
Acknowledgement This work was supported by the National Natural Science Foundation of China (No.61773014), the Postgraduate Research and Practice Innovation Program of Jiangsu Province (No.KYCX19 0250), the Natural Science Foundation of Jiangsu Province (No.BK20160788) and the Fundamental Research Funds for the Central Universities (No.NS2017050).
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