对多重(e,d,N)-策略休假的M_M_c排队的分析
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XU Xiu-li, TIAN Nai-shuo
(The College of Science, Yanshan University, Hebei Qinhuangdao 066004) Abstract: Considered in this paper is a Markovian multi-server queue with the (e, d, N )-policy. By using the quasi birth-and-death process method, we obtain the stationary probability distribution of queue length and LST of conditional waiting time of a customer. Some conditional stochastic decomposition results are also derived. Keywords: (e, d, N )-policy; quasi birth-and-death process; matrix-geometric solution; conditional stochastic decomposition Classification: AMS(2000) 60K25 CLC number: O226 Document code: A
Dec. 2006
CHINESE JOURNAL OF ENGINEERING MATHEMATICS
Vol. 23 No. 6
Article ID:1005-3085(2006)06-1095-06
Analysis on M/M/c Queue with Multiple (e, d, N )-policy Vacation∗
c ≤ k ≤ N − 1.
1097
kµ, (c − d + 1)µ , (c − d + 1)µ kµ 0 , Bk = 0 kµ ( c − e ) µ 0 , 0 kµ ( c − e ) µ 0 , 0 cµ
. CN −1 A B C A .. C .. ..
AN −1 B
.
.
−(λ + kµ), −(λ + kµ) 0 , 0 − ( λ + kµ ) Ak = −(λ + (c − e)µ) 0 , 0 − ( λ + kµ ) −(λ + (c − e)µ) 0 , 0 −(λ + cµ)
1
Introduction
The M/M/c queue with exponentially distributed vacations has been studied by many authors[1−5] . Recently, Zhang and Tian[6−7] studied the vacation models with a “partial server multiple vacation policy” in which some servers (not all) take multiple vacations. In the above literature, the authors have proved several conditional stochastic decomposition results about the queue length and waiting time of a customer. To take over the expenses of switch from taking vacation to serving, and to avoid switching frequently, we introduce threshold value N into the M/M/c queue. In this system, if d servers become idle at the service completion instant, e of d servers take vacation simultaneously, d − e servers stay idle. These e vacation servers return to serve the customer until they find other c − e servers are busy and there are at least N customers in system at some vacation completion instant. This (e, d, N )-policy can be regard as assembly between the M/M/c queue and the M/M/c − e queue. If there are less customers, the queue is a M/M/c − e queue, the other e servers can take assistant work (vacation) to add benefit; If the number of customer in the queue increase to a certain extent, the system revert to the M/M/c queue. The difference between the “(e, d, N )-policy” and the one in [8] is that, in our model, the number of idle servers taking vacations can be adjusted from 1 to d. Specially, the policy is the same as that in [8] if e equals to d. Our model is thus a generalization of the model in [8]. This policy provides more flexibility for the optimization design of the system.
Note that a customer departuring on state (c − d + 1, 1) makes a state transition from (c − d + 1, 1) to (c − d, 0) and all e vacation servers start a vacation at this instant. Using the lexicographical sequence for the states, the infinitesimal generator for the quasi birth-death process is A 0 B1 Q= where C0 A1 B2 C1 A2 .. . C2 .. . BN −1 .. , .
0≤k ≤c−d c − d < k ≤ c − e,
c − e < k ≤ c,
c + 1 ≤ k ≤ N − 1,
万方数据
NO. 6
Xu Xiuli, Tian Naishuo: Analysis on M/M/c Queue with Multiple (e, d, N )-policy Vacation
Biography: Xu Xiuli (Born in 1976), Female, Doctor, lecturer, Research field: Queueing Theory. ∗ Foundation item: The National Natural Science Foundation of China (10271102). Received date: 2004-12-13.
1 ≤ k ≤ c − d, k = c − d + 1, Ck =
λ,
0 ≤ k < c − d, k = c − d, c − d < k ≤ N − 1.
c − d + 1 < k ≤ c − e,
来自百度文库
c − e < k ≤ c − 1,
[λ 0], λ 0 , 0 λ
万方数据
1096 2
CHINESE JOURNAL OF ENGINEERING MATHEMATICS
VOL. 23
Model Description
Consider an M/M/c queue with arrival rate λ and service rate µ for each of c servers. In this system, we introduce the (e, d, N ) vacation policy. The vacation policy prescribes that at a service completion instant, if any d(1 ≤ d < c) servers become idle, e(1 ≤ e < d) of these d servers will take a vacation together. These vacation servers return to serve customers if there are at least N (N ≥ c) customers in system at a vacation termination instant. While these d servers take vacations, the other c − e servers do not take the vacation even they are idle, which means that c − e servers are always available. The vacation time is assumed to be exponentially distributed with the mean being 1/θ. The serve order is First Come First Served (FCFS). In addition, inter-arrival times, service times, and vacation times are mutually independent. Let L(t) be the number of customers in system at time t and 0, there are e vacation servers on vacation at time t, J (t) = 1, there are no servers on vacation at time t. Then {L(t), J (t)} is a quasi-birth-death process (QBD) with the state space Ω = (k, j ) : k ≥ 0, j = 0, 1 .
(The College of Science, Yanshan University, Hebei Qinhuangdao 066004) Abstract: Considered in this paper is a Markovian multi-server queue with the (e, d, N )-policy. By using the quasi birth-and-death process method, we obtain the stationary probability distribution of queue length and LST of conditional waiting time of a customer. Some conditional stochastic decomposition results are also derived. Keywords: (e, d, N )-policy; quasi birth-and-death process; matrix-geometric solution; conditional stochastic decomposition Classification: AMS(2000) 60K25 CLC number: O226 Document code: A
Dec. 2006
CHINESE JOURNAL OF ENGINEERING MATHEMATICS
Vol. 23 No. 6
Article ID:1005-3085(2006)06-1095-06
Analysis on M/M/c Queue with Multiple (e, d, N )-policy Vacation∗
c ≤ k ≤ N − 1.
1097
kµ, (c − d + 1)µ , (c − d + 1)µ kµ 0 , Bk = 0 kµ ( c − e ) µ 0 , 0 kµ ( c − e ) µ 0 , 0 cµ
. CN −1 A B C A .. C .. ..
AN −1 B
.
.
−(λ + kµ), −(λ + kµ) 0 , 0 − ( λ + kµ ) Ak = −(λ + (c − e)µ) 0 , 0 − ( λ + kµ ) −(λ + (c − e)µ) 0 , 0 −(λ + cµ)
1
Introduction
The M/M/c queue with exponentially distributed vacations has been studied by many authors[1−5] . Recently, Zhang and Tian[6−7] studied the vacation models with a “partial server multiple vacation policy” in which some servers (not all) take multiple vacations. In the above literature, the authors have proved several conditional stochastic decomposition results about the queue length and waiting time of a customer. To take over the expenses of switch from taking vacation to serving, and to avoid switching frequently, we introduce threshold value N into the M/M/c queue. In this system, if d servers become idle at the service completion instant, e of d servers take vacation simultaneously, d − e servers stay idle. These e vacation servers return to serve the customer until they find other c − e servers are busy and there are at least N customers in system at some vacation completion instant. This (e, d, N )-policy can be regard as assembly between the M/M/c queue and the M/M/c − e queue. If there are less customers, the queue is a M/M/c − e queue, the other e servers can take assistant work (vacation) to add benefit; If the number of customer in the queue increase to a certain extent, the system revert to the M/M/c queue. The difference between the “(e, d, N )-policy” and the one in [8] is that, in our model, the number of idle servers taking vacations can be adjusted from 1 to d. Specially, the policy is the same as that in [8] if e equals to d. Our model is thus a generalization of the model in [8]. This policy provides more flexibility for the optimization design of the system.
Note that a customer departuring on state (c − d + 1, 1) makes a state transition from (c − d + 1, 1) to (c − d, 0) and all e vacation servers start a vacation at this instant. Using the lexicographical sequence for the states, the infinitesimal generator for the quasi birth-death process is A 0 B1 Q= where C0 A1 B2 C1 A2 .. . C2 .. . BN −1 .. , .
0≤k ≤c−d c − d < k ≤ c − e,
c − e < k ≤ c,
c + 1 ≤ k ≤ N − 1,
万方数据
NO. 6
Xu Xiuli, Tian Naishuo: Analysis on M/M/c Queue with Multiple (e, d, N )-policy Vacation
Biography: Xu Xiuli (Born in 1976), Female, Doctor, lecturer, Research field: Queueing Theory. ∗ Foundation item: The National Natural Science Foundation of China (10271102). Received date: 2004-12-13.
1 ≤ k ≤ c − d, k = c − d + 1, Ck =
λ,
0 ≤ k < c − d, k = c − d, c − d < k ≤ N − 1.
c − d + 1 < k ≤ c − e,
来自百度文库
c − e < k ≤ c − 1,
[λ 0], λ 0 , 0 λ
万方数据
1096 2
CHINESE JOURNAL OF ENGINEERING MATHEMATICS
VOL. 23
Model Description
Consider an M/M/c queue with arrival rate λ and service rate µ for each of c servers. In this system, we introduce the (e, d, N ) vacation policy. The vacation policy prescribes that at a service completion instant, if any d(1 ≤ d < c) servers become idle, e(1 ≤ e < d) of these d servers will take a vacation together. These vacation servers return to serve customers if there are at least N (N ≥ c) customers in system at a vacation termination instant. While these d servers take vacations, the other c − e servers do not take the vacation even they are idle, which means that c − e servers are always available. The vacation time is assumed to be exponentially distributed with the mean being 1/θ. The serve order is First Come First Served (FCFS). In addition, inter-arrival times, service times, and vacation times are mutually independent. Let L(t) be the number of customers in system at time t and 0, there are e vacation servers on vacation at time t, J (t) = 1, there are no servers on vacation at time t. Then {L(t), J (t)} is a quasi-birth-death process (QBD) with the state space Ω = (k, j ) : k ≥ 0, j = 0, 1 .