Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2014, Article ID 738021, 9 pages http://dx.doi.org/10.1155/2014/738021
Research Article Study on the Departure Process of Discrete-Time πΊππ/πΊ/1 Queue with Randomized Vacations Chuanyi Luo,1 Xiaoying Huang,2 and Chuan Ding1 1 2
School of Economical Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China School of Business Administration, Southwestern University of Finance and Economics, Chengdu 611130, China
Correspondence should be addressed to Chuanyi Luo;
[email protected] Received 9 March 2014; Revised 11 May 2014; Accepted 12 May 2014; Published 26 May 2014 Academic Editor: Luca Guerrini Copyright Β© 2014 Chuanyi Luo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper presents an analysis of the departure process of a discrete-time πΊππ/πΊ/1 queue with randomized vacations. By using probability decomposition techniques and renewal process, the expression of expected number of departures during time interval (0+ , π+ ] is derived. The relation among departure process, server state process, and service renewal process is obtained. The relation displays the decomposition characteristic of the departure process. Furthermore, the approximate expansion of the expected number of departures is gained. Since the departure process also often corresponds to an arrival process for a downstream queue in queueing network, it is hoped that the results obtained in this paper may provide useful information for queueing network.
1. Introduction Since discrete-time queues with vacation schedule are more suitable for its applicability in the performance analysis of telecommunication systems, it has gained extended attention. For an excellent survey of earlier works on vacation models as well as its applications, interested readers can refer to [1β 5]. Keilson and Servi (1986) [6] introduced another type of important vacationβBernoulli vacations in which, after each service completion, the server takes a vacation with probability π or continues its busy period (if there are customers in the system) with probability 1 β π. Following the work of Keilson and Servi, much further research on queueing system with Bernoulli vacations was done in [7β10]. On this base, Ke and Chu (2006) [11] develop an interesting vacation discipline from actual situation, where the server takes at most π½ vacations repeatedly until at least one customer is found waiting in the queue upon the server returns from a vacation. In the subsequent research of [12β14], Ke and Huang extend the vacation discipline to the randomized vacation policy with at most π½ vacations where the server takes another vacation with probability π or remains dormant within the system with probability 1 β π if no customer is found waiting in the queue while the server returns from a vacation; otherwise, the server starts to serve the customers
immediately if there are some customers waiting for service at the end of the vacation. This pattern does not terminate until the server has taken π½ successive vacations. Wang (2010) [15] and Wang et al. (2011) [16] introduce the randomized vacation policy to the discrete-time πΊππ/πΊ/1 queue. Such a modified vacation discipline has potentially applications in practical systems [13], for example, in some stochastic production and inventory control systems such as production to orders. This is a continuation of work by Wang et al.(2011) [16] and Luo et al. (2013) [17] where they study the queue size distribution for πΊππ/πΊ/1 with randomized vacation and at most π½ vacations by using different methods, respectively. Instead of studying the queue size distribution studied in [16, 17], this paper considers another themeβthe departure process in that discrete-time model. The investigation of departure process in a queueing system is primarily motivated by the need to analyze queueing network models, in which the departure process of an upstream queue is the arrival process of the downstream queue. Burke (1956) [18] has proven that the departure process of a π/π/π queue is a Bernoulli process. However, it is also well known that if one goes beyond the exponential assumption, unfortunately, the departure process does not become renewal and no exact results of departure process are known for FCFS models. Motivated by the applications of queueing networks
2 in manufacturing and communications, most of researchers adopt approximate method; for example, Whitt (1983) [19] proposes a two-moment theory by approximating the process involved by renewal processes; Harrison and Dai (1989) [20] approximate the queueing network model by a Brownian network which they then solve exactly; Zhang et al. (2005) [21] derive the departure process approximations via an exact aggregate solution technique (called ETAQA); Ferng and Chang (2000) [22] obtain the factorial moments and lag π covariance of interdeparture times by proposing a matrixanalytic approach; and so forth. As far as we know, most of the previous works characterize the departure process by paying close attention to interdeparture times. In this paper, being different from the previous works, we aim at the transient departure number during an arbitrary time interval (0+ , π+ ]. Our objective is to present a distinctive analysis of departure process for the discrete-time queue, obtain the transient expression (π§-transformation) for expected number of departures during the interval (0+ , π+ ], and subsequently discover the decomposition characteristic of the departure process. The remainder of the paper is organized as follows. Section 2 presents the description of the model. Section 3 derives the transient expression for the server busy probability at arbitrary epoch π+ by using π§-transformation. Section 4 gives the transient expression (π§-transformation) for the expected number of departures during the arbitrary interval (0+ , π+ ], denoted by ππ (π+ ). Section 5 gains the approximate expansion of ππ (π+ ). Finally, conclusions are given in Section 7.
2. Model Description Consider a discrete-time πΊππ/πΊ/1 queue with randomized vacations. It is assumed that a potential customer arrives in system during time interval (πβ , π) and a potential departure takes place during time interval (π, π+ ) (LAS-DA) and the input of customers is a Bernoulli process with parameter π (0 < π < 1). Denoting the interarrival time by π, then π{π = π} = π(1 β π)πβ1 , π β₯ 1. Customers are served according to FCFS discipline and the service times for an accepted customer, denoted by π, are independent and identically distributed (i.i.d.) random variables with common probability mass function (p.m.f.) ππ = π{π = π}, π β₯ 1, π probability generating function (P.G.F.) πΊ(π§) = ββ π=1 ππ π§ , and mean service time πΈ[π] = πΌ. After all the customers in the queue are served exhaustively, the server operates a randomized vacation policy with at most π½ vacations. As soon as the system becomes empty, the server immediately takes a vacation, denoted by π, with p.m.f. Vπ = π{π = π}, π β₯ 1 and P.G.F. V(π§). If there is no customer in the system when the server returns from the vacation, the server takes another vacation with probability π or remains dormant within the system with probability π = 1 β π. Otherwise, the server starts to serve the customers immediately if there are some customers waiting for service at the end of vacation. This pattern repeats until the server has taken π½ successive vacations. If the system remains empty at the end of the π½th
Discrete Dynamics in Nature and Society Arrival
nβ
Arrival
n+
n
β
(n + 1)β
Departure
n+1
(n + 1)+
Departure
Potential departure epoch Potential beginning or end Potential arrival epoch of service and vacation β Outside observerβs observation epoch
Figure 1: Various time epochs in a late arrival system with delayed access (LAS-DA).
vacation, the server keeps idle in the system until a next customer arrives, who evokes immediately service for the arrival. Furthermore, various stochastic processes involved in the system are assumed to be mutually independent. To make it clear, the various time epochs at which events occur are shown in a self-explanatory figure (see Figure 1).
3. Server Busy Probability at an Arbitrary Epoch π+ Denoting by π the length of server busy period evoked by only one customer with P.G.F. π΅(π§), then it follows the lemma which is also obtained by Bruneel and Kim (1993) [1]. Lemma 1. In πΊππ/πΊ/1 queue, for |π§| < 1, π΅(π§) is the root of the following equation: π΅ (π§) = πΊ [π§ β π§π (1 β π΅ (π§))] , πΌ , π < 1, πΈ [π] = 1βπ
(1)
where π = ππΌ. Let π be the length of server busy period evoked by π customers; thus π can be expressed as π = βπV=1 πV , where π1 , π2 , . . . , ππ are independent of each other and have the identical distribution as π. So the P.G.F. of π is given by π΅π (π§). Denote by π(π+ ) the queue length at arbitrary epoch π+ . Let π(π+ ) = 1 be the server busy state; that is, the server is busy at epoch π+ . Define π΄ π (π+ ) = π{π(π+ ) = 1 | π(0+ ) = π} with + π π§-transform ππ (π’) = ββ π=0 π΄ π (π )π’ , |π’| < 1; thus the following expression of ππ (π’) holds. Theorem 2. In πΊππ/πΊ/1 queue with randomized vacations, for |π’| < 1 and π β₯ 1, one has π½
π0 (π’) = ((1 β ππ’) [1 β (πV (ππ’)) ] Γ [V (π’) β V (ππ’π΅ (π’) + ππ’)] +ππ’ (1 β π΅ (π’)) β
π¦ (π’) ) β1
Γ ((1 β π’) [(1 β ππ’) β
π₯ (π’) β ππ’π΅ (π’) β
π¦ (π’)]) , ππ (π’) =
1 β π΅π (π’) + π΅π (π’) β
π0 (π’) , 1βπ’
(2)
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3
and conditioning on π < 1, the steady state probability is given by lim π΄ 0 (π+ ) = lim π΄ π (π+ ) = π,
πββ
(3)
πββ
where
customers arriving during the time interval [π, π ]. As a result, the first item of (6) is equal to π½
β
π=1
π=1
βππβ1 β π {π (π+ ) = 1; π β€ π+ ; π < π; π + ππβ1 β€ π < π + ππ }
π₯ (π’) = 1 β πV (ππ’) π½
π½
β
π=1
π=1
= βππβ1 β π {π (π+ ) = 1; π β€ π+ ; π < π;
β [1 β (πV (ππ’)) ] [V (ππ’π΅ (π’) + ππ’) β V (ππ’)] π½
π¦ (π’) = (1 β π) V (ππ’) + [πV (ππ’)] [1 β V (ππ’)] .
(π β π ) + ππβ1 β€ π
(4) Proof. In consideration that the input process is Bernoulli process, the ending points of a server busy period and a vacation are all renewal points and the system is going between server busy state and unoccupied state (vacation state or potential server idle state). To derive the expressions of ππ (π’), the following notations are introduced: π
ππ = β ππ , π=1
π
π
= β ππ ,
(7)
< (π β π ) + ππ } π½
β
π
= βππβ1 β β π {π = π} π=1
π=1π=πβ1
πβπ
πβπ π Γ β π {π = π} β
π {π > π} ( ) ππ π π π=1
β
π΄ π ((π β π β π)+ ) .
π0 = 0, (5) π
= 0,
π=1
where π1 , π2 , . . . , ππ denote the interarrival times of the Bernoulli process with rate π. π1 , π2 , . . . , ππ represent the vacations which are i.i.d. random variables with the same distribution as π. By using renewal process theory and techniques of probability decomposition, it gets
One should note that the item of β(π β π )β in the above equation represents the remaining interarrival time at the ending epoch of the (π β 1)th vacation. Based on the βlack of memory propertyβ of the Bernoulli process, the β(πβπ )β has the same distribution as π under the condition of π = π (π = π β 1, π, . . .). The second item of (6) is given by π½
π
π=2
π=1
βππβ2 (1 β π) β π {π = π} π {π < π} β
π΄ 1 ((π β π)+ ) .
+
π΄ 0 (π ) = π {π (π+ ) = 1 | π (0+ ) = 0} = π {π β€ π+ ; π (π+ ) = 1} π½
(8) Multiplying (6) by π’π and summing over π after substituting (7) and (8) into (6) gain
= βπ {π β€ π+ ; π (π+ ) = 1; π < π β€ π } π=1
+ π {π (π+ ) = 1; π β€ π+ ; π < π} π½
= βππβ1 π {π (π+ ) = 1; π β€ π+ ; π < π β€ π } π=1
π0 (π’) =
1 β [πV (ππ’)]
π½
1 β πV (ππ’) β π
π½
+ βππβ2 (1 β π) π {π (π+ ) = 1; π β€ π+ ; π < π}
πβπ π β
β βπ {π = π} π’π ( ) ππ π ππ (π’) π π=1 π=1
π=2
+ π1 (π’) β
π
+ ππ½β1 βπ {π = π} π {π < π} β
π΄ 1 ((π β π)+ ) . π=1
(6)
+
In the first item of (6), the βπ β€ π ; π < π β€ π β means that the epoch π locates behind the πth vacation and the first customer arrives in system during the πth vacation. So there would be some other potential
β
ππ’ (1 β π) V (ππ’)
(9)
1 β ππ’
1 β [πV (ππ’)]
π½β1
1 β πV (ππ’)
+ π1 (π’) β
ππ’ β
ππ½β1 [V (ππ’)] 1 β ππ’
π½
.
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Discrete Dynamics in Nature and Society
For 1 β€ π, it gets
where πΊ(π’) is the P.G.F. of distribution of service time {ππ , π β₯ 1} and πΌ = πΈ[ππ ].
π΄ π (π+ ) = π {π (π+ ) = 1; π > π+ } +
+ π {π (π ) = 1; π
Proof. One has
π
+
β€ π } = 1 β βπ {π
= π}
π=π
π=0
π
+ βπ {π π=π
+
= π} π΄ 0 ((π β π) ) .
β
(10)
ππ (π’) =
1 β π΅π (π’) + π΅π (π’) β
π0 (π’) . 1βπ’
(11)
Remark 3. By assuming π{π = 0} = 1, the model considered here becomes a special case where the vacation disappears. We can easily derive the corresponding expression of ππ (π’) in classical πΊππ/πΊ/1 queueing model without vacation policy. Consider the following: ππ’ (1 β π΅ (π’)) (1 β π’) [(1 β ππ’) β ππ’π΅ (π’)]
(12)
One may note here that the steady state of π΄ 0 (π+ ) has nothing to do with the vacation policy.
(13)
Thus, π·(π+ ) is a renewal process and denotes the number of departures during time interval (0+ , π+ ] contained in the service process {ππ , π β₯ 1}. Let π(π+ ) = πΈ[π·(π+ )] be the renewal function of π·(π+ ) with π§-transform π(π’) = + π ββ π=0 π(π )π’ . Lemma 4. For |π’| < 1, it has π (π+ ) 1 = , (14) πββ π πΌ lim
π· (π+ ) = {The number of departures during
ππ (π+ ) = πΈ [π· (π+ ) | π (0+ ) = π] ,
(17)
π, π β₯ 0.
ππ (π+ ) = πΈ {The number of departures during (0+ , π ] ; π β€ π+ } ,
β
π‘π (π’) = β ππ (π+ ) π’π , π=0
π· (π+ ) = sup {π : ππ β€ π+ } .
πΊ (π’) , (1 β π’) [1 β πΊ (π’)]
(16)
On this basis, we begin to consider the expected number of departures during an arbitrary time interval (0+ , π+ ] in the queue system. For this aim, some additional notations are developed as follows:
(0+ , π+ ] ; π > π+ } ,
Consider a renewal process driven by a list of service times {ππ , π β₯ 1} defined in Section 1. Let
π=1
π (π+ ) 1 1 = . = πββ π πΈ [ππ ] πΌ lim
ππ (π+ ) = πΈ {The number of departures during
4. Expected Number of Departures during the Arbitrary Time Interval (0+ , π+ ]
π0 = 0,
Taking π§-transform on both sides of the above equation yields the expression of π(π’). Applying the essential renewal theorem (see [24]), it gets
So, ππ (π+ ) represents the expected departures during (0+ , π+ ] with the initial state of π(0+ ) = π. One has the following:
πββ
π
(15)
general time interval (0+ , π+ ]} ,
lim π΄ 0 (π+ ) = lim π΄ π (π+ ) = π.
ππ = βππ ,
π=0π=π
,
1 β π΅π (π’) ππ (π’) = + π΅π (π’) β
π0 (π’) , 1βπ’ πββ
π=0
= β β ππ {π1 + β
β
β
+ ππ = π} π {ππ+1 > (π β π)+ } .
Solving the simultaneous equations (9) and (11) leads to the expression of ππ (π’) provided in Theorem 2. Applying the Final Value Theorem (see [23]), it has π΄ (π+ ) = lim (1 β π’) β
ππ (π’). By using Lemma 1, lim πββ π π’ β 1β + the stable result of π΄ π (π ) given in Theorem 2 is obtained.
π0 (π’) =
= β ππ {π1 + β
β
β
+ ππ β€ π+ < π1 + β
β
β
+ ππ+1 } β π
Multiplying (10) by π’π and summing over π yield
π (π’) =
β
π (π+ ) = πΈ [π· (π+ )] = β ππ {π· (π+ ) = π}
(18)
π = 1, 2, . . . , β
π€π (π’) = β ππ (π+ ) π’π , π=0
where π denotes the length of a server busy period evoked by π customers waiting for service. + π Theorem 5. Let ππ (π’) = ββ π=0 ππ (π )π’ , for |π’| < 1, π < 1, and π β₯ 0; it has
ππ (π’) = π (π’) β
(1 β π’) ππ (π’) , π, π < 0 { ππ (π+ ) { ={ πββ {1 π , π β₯ 1, {πΌ
(19)
lim
where ππ (π’) and π(π’) are determined by Theorem 2 and Lemma 4, respectively.
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Proof. Since the arrival process is Bernoulli process, the beginning and ending epochs of a server busy period or a vacation are both renewal points. Thus, it can take probability decomposition techniques on π(π+ ) by using π . Consider π (π+ ) = πΈ [π· (π+ )] = πΈ [π· (π+ ) ; π > π+ ] + πΈ [π· (π+ ) ; π β€ π+ ]
Multiplying (22) by π’π and summing over π result in
π0 (π’) =
1 β [πV (ππ’)]
π½
1 β πV (ππ’) β π
πβπ π β
β βπ {π = π} π’π ( ) ππ π ππ (π’) π π=1
= ππ (π+ ) + πΈ [π· (π ) ; π β€ π+ ]
π=1
π
+ π1 (π’) β
+ βπ {π = π} πΈ [π· ((π β π)+ )]
ππ’ (1 β π) V (ππ’) 1 β ππ’
π=π π
β
= ππ (π+ ) + ππ (π+ ) + βπ {π = π} π ((π β π)+ ) . π=π
(20) Taking π§-transform on both sides of the above equation and applying Lemma 4 give π‘π (π’) + π€π (π’) =
[1 β π΅π (π’)] πΊ (π’) (1 β π’) [1 β πΊ (π’)]
.
(21)
For π0 (π+ ), by using the same method used for the solution of ππ (π’) (π β₯ 0), it gets π0 (π+ ) = πΈ [π· (π+ ) | π (0+ ) = 0] = πΈ [π· (π+ ) ; π β€ π+ ] π½
= βπΈ [π· (π+ ) ; π β€ π+ ; π < π β€ π ]
β
1 β [πV (ππ’)]
(23)
π½β1
+ π1 (π’)
1 β πV (ππ’) ππ’ β
ππ½β1 [V (ππ’)] 1 β ππ’
π½
.
For ππ (π+ ) (π β₯ 1), it has ππ (π+ ) = πΈ [π· (π+ ) | π (0+ ) = π] = πΈ [π· (π+ ) ; π+ < π ] + πΈ [π· (π+ ) ; π β€ π+ ] = ππ (π+ ) + πΈ {The number of departures during
π=1
(0+ , π ] ; π β€ π+ }
+ πΈ [π· (π+ ) ; π β€ π+ ; π < π] π½
= βππβ1 πΈ [π· (π+ ) ; π β€ π+ ; π < π β€ π ] π=1
+ πΈ {The number of departures during (π , π+ ] ; π β€ π+ }
π½
+ βππβ2 (1 β π) πΈ [π· (π+ ) ; π β€ π+ ; π < π] π=2
π
= ππ (π+ ) + ππ (π+ ) + βπ {π = π} π0 ((π β π)+ ) . π=π
π
+ ππ½β1 βπ {π = π} π {π < π} β
π1 ((π β π)+ )
(24)
π=1
π½
β
π
= βππβ1 β β π {π = π}
Taking π§-transform on both sides of (24) and applying (21) yield
π=1π=πβ1
π=1 πβπ
πβπ π Γ β π {π = π} β
π {π > π} ( ) ππ π π π=1
+
π½
πβ2
β
ππ ((π β π β π) ) + βπ
ππ (π’) = (1 β π)
[1 β π΅π (π’)] πΊ (π’) (1 β π’) [1 β πΊ (π’)]
+ π΅π (π’) π0 (π’) .
(25)
π=2
π
Solving the simultaneous equations (23) and (25), it gets the expression of ππ (π’) (π β₯ 0). From the expressions of ππ (π’) (π β₯ 0) given here, ππ (π’) (π β₯ 0) given by Theorem 2, and π(π’) given by Lemma 4, the relation among ππ (π’), π(π’), and ππ (π’) is established as follows:
Γ β π {π = π} π {π < π} π=1
β
π1 ((π β π)+ ) + ππ½β1 π
Γ βπ {π = π} π {π < π} β
π1 ((π β π)+ ) . π=1
(22)
ππ (π’) = π (π’) β
(1 β π’) ππ (π’) .
(26)
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Discrete Dynamics in Nature and Society
Applying Theorem 2 and Lemma 4, when condition π < 1 holds, it has the following steady result: ππ (π+ ) = limβ (1 β π’)2 ππ (π’) πββ π’β1 π lim
= limβ (1 β π’)2 π (π’) β
limβ (1 β π’) ππ (π’) π’β1
π’β1
+
)+1 ππ ; thus ππ·(π+ )+1 denotes the Proof. Let ππ·(π+ )+1 = βπ·(π π=1 epoch immediately after the (π·(π+ )+1)th service completion. + Denote by π(π ) the remain service time of a customer being served at epoch π+ with the corresponding steady state π+ = + lim π(π ) and the steady state distribution π₯π = π{π+ = πββ
π π}, π β₯ 1, which has P.G.F. πβ (π’) = ββ π=1 π₯π π’ . From [25], it has
π (π+ ) β
lim π΄ (π+ ) πββ πββ π
= lim
1 { β
π = π, { {πΌ ={ {1 { β
1, {πΌ
πβ (π’) =
π