Research Article Study on the Departure Process of Discrete ... - Hindawi

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May 12, 2014 - Chuanyi Luo,1 Xiaoying Huang,2 and Chuan Ding1. 1 School of Economical Mathematics, Southwestern University of Finance and Economics,Β ...
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 πΊπ‘’π‘œ/𝐺/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 πΊπ‘’π‘œ/𝐺/1 queue with randomized vacations. By using probability decomposition techniques and renewal process, the expression of expected number of departures during time interval (0+ , 𝑛+ ] 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 [1– 5]. Keilson and Servi (1986) [6] introduced another type of important vacationβ€”Bernoulli vacations in which, after each service completion, the server takes a vacation with probability 𝑝 or continues its busy period (if there are customers in the system) with probability 1 βˆ’ 𝑝. Following the work of Keilson and Servi, much further research on queueing system with Bernoulli vacations was done in [7–10]. On this base, Ke and Chu (2006) [11] develop an interesting vacation discipline from actual situation, where the server takes at most 𝐽 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 [12–14], Ke and Huang extend the vacation discipline to the randomized vacation policy with at most 𝐽 vacations where the server takes another vacation with probability 𝑝 or remains dormant within the system with probability 1 βˆ’ 𝑝 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 𝐽 successive vacations. Wang (2010) [15] and Wang et al. (2011) [16] introduce the randomized vacation policy to the discrete-time πΊπ‘’π‘œ/𝐺/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 πΊπ‘’π‘œ/𝐺/1 with randomized vacation and at most 𝐽 vacations by using different methods, respectively. Instead of studying the queue size distribution studied in [16, 17], this paper considers another themeβ€”the 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 𝑀/𝑀/𝑠 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 𝑛 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+ , 𝑛+ ]. Our objective is to present a distinctive analysis of departure process for the discrete-time queue, obtain the transient expression (𝑧-transformation) for expected number of departures during the interval (0+ , 𝑛+ ], 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 𝑛+ by using 𝑧-transformation. Section 4 gives the transient expression (𝑧-transformation) for the expected number of departures during the arbitrary interval (0+ , 𝑛+ ], denoted by 𝑀𝑖 (𝑛+ ). Section 5 gains the approximate expansion of 𝑀𝑖 (𝑛+ ). Finally, conclusions are given in Section 7.

2. Model Description Consider a discrete-time πΊπ‘’π‘œ/𝐺/1 queue with randomized vacations. It is assumed that a potential customer arrives in system during time interval (π‘›βˆ’ , 𝑛) and a potential departure takes place during time interval (𝑛, 𝑛+ ) (LAS-DA) and the input of customers is a Bernoulli process with parameter πœ† (0 < πœ† < 1). Denoting the interarrival time by 𝜏, then 𝑃{𝜏 = 𝑗} = πœ†(1 βˆ’ πœ†)π‘—βˆ’1 , 𝑗 β‰₯ 1. Customers are served according to FCFS discipline and the service times for an accepted customer, denoted by πœ’, are independent and identically distributed (i.i.d.) random variables with common probability mass function (p.m.f.) 𝑔𝑗 = 𝑃{πœ’ = 𝑗}, 𝑗 β‰₯ 1, 𝑗 probability generating function (P.G.F.) 𝐺(𝑧) = βˆ‘βˆž 𝑗=1 𝑔𝑗 𝑧 , and mean service time 𝐸[πœ’] = 𝛼. After all the customers in the queue are served exhaustively, the server operates a randomized vacation policy with at most 𝐽 vacations. As soon as the system becomes empty, the server immediately takes a vacation, denoted by 𝑉, with p.m.f. V𝑗 = 𝑃{𝑉 = 𝑗}, 𝑗 β‰₯ 1 and P.G.F. V(𝑧). If there is no customer in the system when the server returns from the vacation, the server takes another vacation with probability πœƒ or remains dormant within the system with probability πœƒ = 1 βˆ’ πœƒ. 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 𝐽 successive vacations. If the system remains empty at the end of the 𝐽th

Discrete Dynamics in Nature and Society Arrival

nβˆ’

Arrival

n+

n

βˆ—

(n + 1)βˆ’

Departure

n+1

(n + 1)+

Departure

Potential departure epoch Potential beginning or end Potential arrival epoch of service and vacation βˆ— Outside observer’s 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 𝑛+ Denoting by 𝑏 the length of server busy period evoked by only one customer with P.G.F. 𝐡(𝑧), then it follows the lemma which is also obtained by Bruneel and Kim (1993) [1]. Lemma 1. In πΊπ‘’π‘œ/𝐺/1 queue, for |𝑧| < 1, 𝐡(𝑧) is the root of the following equation: 𝐡 (𝑧) = 𝐺 [𝑧 βˆ’ π‘§πœ† (1 βˆ’ 𝐡 (𝑧))] , 𝛼 , 𝜌 < 1, 𝐸 [𝑏] = 1βˆ’πœŒ

(1)

where 𝜌 = πœ†π›Ό. Let 𝑏 be the length of server busy period evoked by 𝑖 customers; thus 𝑏 can be expressed as 𝑏 = βˆ‘π‘–V=1 𝑏V , where 𝑏1 , 𝑏2 , . . . , 𝑏𝑖 are independent of each other and have the identical distribution as 𝑏. So the P.G.F. of 𝑏 is given by 𝐡𝑖 (𝑧). Denote by 𝑁(𝑛+ ) the queue length at arbitrary epoch 𝑛+ . Let πœ‰(𝑛+ ) = 1 be the server busy state; that is, the server is busy at epoch 𝑛+ . Define 𝐴 𝑖 (𝑛+ ) = 𝑃{πœ‰(𝑛+ ) = 1 | 𝑁(0+ ) = 𝑖} with + 𝑛 𝑧-transform π‘Žπ‘– (𝑒) = βˆ‘βˆž 𝑛=0 𝐴 𝑖 (𝑛 )𝑒 , |𝑒| < 1; thus the following expression of π‘Žπ‘– (𝑒) holds. Theorem 2. In πΊπ‘’π‘œ/𝐺/1 queue with randomized vacations, for |𝑒| < 1 and 𝑖 β‰₯ 1, one has 𝐽

π‘Ž0 (𝑒) = ((1 βˆ’ πœ†π‘’) [1 βˆ’ (πœƒV (πœ†π‘’)) ] Γ— [V (𝑒) βˆ’ V (πœ†π‘’π΅ (𝑒) + πœ†π‘’)] +πœ†π‘’ (1 βˆ’ 𝐡 (𝑒)) β‹… 𝑦 (𝑒) ) βˆ’1

Γ— ((1 βˆ’ 𝑒) [(1 βˆ’ πœ†π‘’) β‹… π‘₯ (𝑒) βˆ’ πœ†π‘’π΅ (𝑒) β‹… 𝑦 (𝑒)]) , π‘Žπ‘– (𝑒) =

1 βˆ’ 𝐡𝑖 (𝑒) + 𝐡𝑖 (𝑒) β‹… π‘Ž0 (𝑒) , 1βˆ’π‘’

(2)

Discrete Dynamics in Nature and Society

3

and conditioning on 𝜌 < 1, the steady state probability is given by lim 𝐴 0 (𝑛+ ) = lim 𝐴 𝑖 (𝑛+ ) = 𝜌,

π‘›β†’βˆž

(3)

π‘›β†’βˆž

where

customers arriving during the time interval [𝜏, 𝑉 ]. As a result, the first item of (6) is equal to 𝐽

∞

𝑖=1

π‘š=1

βˆ‘πœƒπ‘–βˆ’1 βˆ‘ 𝑃 {πœ‰ (𝑛+ ) = 1; 𝑉 ≀ 𝑛+ ; 𝑉 < 𝜏; 𝜏 + π‘†π‘šβˆ’1 ≀ 𝑉 < 𝜏 + π‘†π‘š }

π‘₯ (𝑒) = 1 βˆ’ πœƒV (πœ†π‘’) 𝐽

𝐽

∞

𝑖=1

π‘š=1

= βˆ‘πœƒπ‘–βˆ’1 βˆ‘ 𝑃 {πœ‰ (𝑛+ ) = 1; 𝑉 ≀ 𝑛+ ; 𝑉 < 𝜏;

βˆ’ [1 βˆ’ (πœƒV (πœ†π‘’)) ] [V (πœ†π‘’π΅ (𝑒) + πœ†π‘’) βˆ’ V (πœ†π‘’)] 𝐽

𝑦 (𝑒) = (1 βˆ’ πœƒ) V (πœ†π‘’) + [πœƒV (πœ†π‘’)] [1 βˆ’ V (πœ†π‘’)] .

(𝜏 βˆ’ 𝑉 ) + π‘†π‘šβˆ’1 ≀ 𝑉

(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 π‘Žπ‘– (𝑒), the following notations are introduced: π‘š

π‘†π‘š = βˆ‘ πœπ‘˜ , π‘˜=1

𝑉



𝑖

= βˆ‘ π‘‰π‘˜ ,

(7)

< (𝜏 βˆ’ 𝑉 ) + π‘†π‘š } 𝐽

∞

𝑛

= βˆ‘πœƒπ‘–βˆ’1 βˆ‘ βˆ‘ 𝑃 {𝑉 = π‘˜} 𝑖=1

π‘š=1π‘˜=π‘–βˆ’1

π‘›βˆ’π‘˜

π‘™βˆ’π‘š 𝑙 Γ— βˆ‘ 𝑃 {𝑉 = 𝑙} β‹… 𝑃 {𝜏 > π‘˜} ( ) πœ†π‘š πœ† π‘š 𝑙=1

β‹… 𝐴 π‘š ((𝑛 βˆ’ π‘˜ βˆ’ 𝑙)+ ) .

𝑆0 = 0, (5) 𝑉



= 0,

π‘˜=1

where 𝜏1 , 𝜏2 , . . . , πœπ‘š denote the interarrival times of the Bernoulli process with rate πœ†. 𝑉1 , 𝑉2 , . . . , 𝑉𝑖 represent the vacations which are i.i.d. random variables with the same distribution as 𝑉. By using renewal process theory and techniques of probability decomposition, it gets

One should note that the item of β€œ(𝜏 βˆ’ 𝑉 )” in the above equation represents the remaining interarrival time at the ending epoch of the (𝑖 βˆ’ 1)th vacation. Based on the β€œlack of memory property” of the Bernoulli process, the β€œ(πœβˆ’π‘‰ )” has the same distribution as 𝜏 under the condition of 𝑉 = π‘˜ (π‘˜ = 𝑖 βˆ’ 1, 𝑖, . . .). The second item of (6) is given by 𝐽

𝑛

𝑖=2

π‘˜=1

βˆ‘πœƒπ‘–βˆ’2 (1 βˆ’ πœƒ) βˆ‘ 𝑃 {𝜏 = π‘˜} 𝑃 {𝑉 < π‘˜} β‹… 𝐴 1 ((𝑛 βˆ’ π‘˜)+ ) .

+

𝐴 0 (𝑛 ) = 𝑃 {πœ‰ (𝑛+ ) = 1 | 𝑁 (0+ ) = 0} = 𝑃 {𝜏 ≀ 𝑛+ ; πœ‰ (𝑛+ ) = 1} 𝐽

(8) Multiplying (6) by 𝑒𝑛 and summing over 𝑛 after substituting (7) and (8) into (6) gain

= βˆ‘π‘ƒ {𝜏 ≀ 𝑛+ ; πœ‰ (𝑛+ ) = 1; 𝑉 < 𝜏 ≀ 𝑉 } 𝑖=1

+ 𝑃 {πœ‰ (𝑛+ ) = 1; 𝜏 ≀ 𝑛+ ; 𝑉 < 𝜏} 𝐽

= βˆ‘πœƒπ‘–βˆ’1 𝑃 {πœ‰ (𝑛+ ) = 1; 𝑉 ≀ 𝑛+ ; 𝑉 < 𝜏 ≀ 𝑉 } 𝑖=1

π‘Ž0 (𝑒) =

1 βˆ’ [πœƒV (πœ†π‘’)]

𝐽

1 βˆ’ πœƒV (πœ†π‘’) ∞ π‘š

𝐽

+ βˆ‘πœƒπ‘–βˆ’2 (1 βˆ’ πœƒ) 𝑃 {πœ‰ (𝑛+ ) = 1; 𝜏 ≀ 𝑛+ ; 𝑉 < 𝜏}

π‘šβˆ’π‘™ π‘š β‹… βˆ‘ βˆ‘π‘ƒ {𝑉 = π‘š} π‘’π‘š ( ) πœ†π‘™ πœ† π‘Žπ‘™ (𝑒) 𝑙 π‘š=1 𝑙=1

𝑖=2

+ π‘Ž1 (𝑒) β‹…

𝑛

+ πœƒπ½βˆ’1 βˆ‘π‘ƒ {𝜏 = 𝑖} 𝑃 {𝑉 < 𝑖} β‹… 𝐴 1 ((𝑛 βˆ’ 𝑖)+ ) . 𝑖=1

(6)

+



In the first item of (6), the β€œπ‘‰ ≀ 𝑛 ; 𝑉 < 𝜏 ≀ 𝑉 ” means that the epoch 𝑛 locates behind the 𝑖th vacation and the first customer arrives in system during the 𝑖th vacation. So there would be some other potential

β‹…

πœ†π‘’ (1 βˆ’ πœƒ) V (πœ†π‘’)

(9)

1 βˆ’ πœ†π‘’

1 βˆ’ [πœƒV (πœ†π‘’)]

π½βˆ’1

1 βˆ’ πœƒV (πœ†π‘’)

+ π‘Ž1 (𝑒) β‹…

πœ†π‘’ β‹… πœƒπ½βˆ’1 [V (πœ†π‘’)] 1 βˆ’ πœ†π‘’

𝐽

.

4

Discrete Dynamics in Nature and Society

For 1 ≀ 𝑖, it gets

where 𝐺(𝑒) is the P.G.F. of distribution of service time {πœ’π‘– , 𝑖 β‰₯ 1} and 𝛼 = 𝐸[πœ’π‘– ].

𝐴 𝑖 (𝑛+ ) = 𝑃 {πœ‰ (𝑛+ ) = 1; 𝑏 > 𝑛+ } +



+ 𝑃 {πœ‰ (𝑛 ) = 1; 𝑏

Proof. One has

𝑛

+



≀ 𝑛 } = 1 βˆ’ βˆ‘π‘ƒ {𝑏

= π‘˜}

π‘˜=𝑖

π‘˜=0

𝑛



+ βˆ‘π‘ƒ {𝑏 π‘˜=𝑖

+

= π‘˜} 𝐴 0 ((𝑛 βˆ’ π‘˜) ) .

∞

(10)

π‘Žπ‘– (𝑒) =

1 βˆ’ 𝐡𝑖 (𝑒) + 𝐡𝑖 (𝑒) β‹… π‘Ž0 (𝑒) . 1βˆ’π‘’

(11)

Remark 3. By assuming 𝑃{𝑉 = 0} = 1, the model considered here becomes a special case where the vacation disappears. We can easily derive the corresponding expression of π‘Žπ‘– (𝑒) in classical πΊπ‘’π‘œ/𝐺/1 queueing model without vacation policy. Consider the following: πœ†π‘’ (1 βˆ’ 𝐡 (𝑒)) (1 βˆ’ 𝑒) [(1 βˆ’ πœ†π‘’) βˆ’ πœ†π‘’π΅ (𝑒)]

(12)

One may note here that the steady state of 𝐴 0 (𝑛+ ) has nothing to do with the vacation policy.

(13)

Thus, 𝐷(𝑛+ ) is a renewal process and denotes the number of departures during time interval (0+ , 𝑛+ ] contained in the service process {πœ’π‘– , 𝑖 β‰₯ 1}. Let 𝑀(𝑛+ ) = 𝐸[𝐷(𝑛+ )] be the renewal function of 𝐷(𝑛+ ) with 𝑧-transform π‘š(𝑒) = + 𝑛 βˆ‘βˆž 𝑛=0 𝑀(𝑛 )𝑒 . Lemma 4. For |𝑒| < 1, it has 𝑀 (𝑛+ ) 1 = , (14) π‘›β†’βˆž 𝑛 𝛼 lim

𝐷 (𝑛+ ) = {The number of departures during

𝑀𝑖 (𝑛+ ) = 𝐸 [𝐷 (𝑛+ ) | 𝑁 (0+ ) = 𝑖] ,

(17)

𝑛, 𝑖 β‰₯ 0.

𝑇𝑖 (𝑛+ ) = 𝐸 {The number of departures during (0+ , 𝑏 ] ; 𝑏 ≀ 𝑛+ } ,

∞

𝑑𝑖 (𝑒) = βˆ‘ 𝑇𝑖 (𝑛+ ) 𝑒𝑛 , 𝑛=0

𝐷 (𝑛+ ) = sup {𝑛 : π‘ˆπ‘› ≀ 𝑛+ } .

𝐺 (𝑒) , (1 βˆ’ 𝑒) [1 βˆ’ 𝐺 (𝑒)]

(16)

On this basis, we begin to consider the expected number of departures during an arbitrary time interval (0+ , 𝑛+ ] in the queue system. For this aim, some additional notations are developed as follows:

(0+ , 𝑛+ ] ; 𝑏 > 𝑛+ } ,

Consider a renewal process driven by a list of service times {πœ’π‘– , 𝑖 β‰₯ 1} defined in Section 1. Let

𝑖=1

𝑀 (𝑛+ ) 1 1 = . = π‘›β†’βˆž 𝑛 𝐸 [πœ’π‘– ] 𝛼 lim

π‘Šπ‘– (𝑛+ ) = 𝐸 {The number of departures during

4. Expected Number of Departures during the Arbitrary Time Interval (0+ , 𝑛+ ]

π‘ˆ0 = 0,

Taking 𝑧-transform on both sides of the above equation yields the expression of π‘š(𝑒). Applying the essential renewal theorem (see [24]), it gets

So, 𝑀𝑖 (𝑛+ ) represents the expected departures during (0+ , 𝑛+ ] with the initial state of 𝑁(0+ ) = 𝑖. One has the following:

π‘›β†’βˆž

𝑛

(15)

general time interval (0+ , 𝑛+ ]} ,

lim 𝐴 0 (𝑛+ ) = lim 𝐴 𝑖 (𝑛+ ) = 𝜌.

π‘ˆπ‘› = βˆ‘πœ’π‘– ,

π‘˜=0π‘š=π‘˜

,

1 βˆ’ 𝐡𝑖 (𝑒) π‘Žπ‘– (𝑒) = + 𝐡𝑖 (𝑒) β‹… π‘Ž0 (𝑒) , 1βˆ’π‘’ π‘›β†’βˆž

π‘˜=0

= βˆ‘ βˆ‘ π‘˜π‘ƒ {πœ’1 + β‹… β‹… β‹… + πœ’π‘˜ = π‘š} 𝑃 {πœ’π‘˜+1 > (𝑛 βˆ’ π‘š)+ } .

Solving the simultaneous equations (9) and (11) leads to the expression of π‘Žπ‘– (𝑒) provided in Theorem 2. Applying the Final Value Theorem (see [23]), it has 𝐴 (𝑛+ ) = lim (1 βˆ’ 𝑒) β‹… π‘Žπ‘– (𝑒). By using Lemma 1, lim π‘›β†’βˆž 𝑖 𝑒 β†’ 1βˆ’ + the stable result of 𝐴 𝑖 (𝑛 ) given in Theorem 2 is obtained.

π‘Ž0 (𝑒) =

= βˆ‘ π‘˜π‘ƒ {πœ’1 + β‹… β‹… β‹… + πœ’π‘˜ ≀ 𝑛+ < πœ’1 + β‹… β‹… β‹… + πœ’π‘˜+1 } ∞ 𝑛

Multiplying (10) by 𝑒𝑛 and summing over 𝑛 yield

π‘š (𝑒) =

∞

𝑀 (𝑛+ ) = 𝐸 [𝐷 (𝑛+ )] = βˆ‘ π‘˜π‘ƒ {𝐷 (𝑛+ ) = π‘˜}

(18)

𝑖 = 1, 2, . . . , ∞

𝑀𝑖 (𝑒) = βˆ‘ π‘Šπ‘– (𝑛+ ) 𝑒𝑛 , 𝑛=0

where 𝑏 denotes the length of a server busy period evoked by 𝑖 customers waiting for service. + 𝑛 Theorem 5. Let π‘šπ‘– (𝑒) = βˆ‘βˆž 𝑛=0 𝑀𝑖 (𝑛 )𝑒 , for |𝑒| < 1, 𝜌 < 1, and 𝑖 β‰₯ 0; it has

π‘šπ‘– (𝑒) = π‘š (𝑒) β‹… (1 βˆ’ 𝑒) π‘Žπ‘– (𝑒) , πœ†, 𝜌 < 0 { 𝑀𝑖 (𝑛+ ) { ={ π‘›β†’βˆž {1 𝑛 , 𝜌 β‰₯ 1, {𝛼

(19)

lim

where π‘Žπ‘– (𝑒) and π‘š(𝑒) are determined by Theorem 2 and Lemma 4, respectively.

Discrete Dynamics in Nature and Society

5

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 𝑀(𝑛+ ) by using 𝑏 . Consider 𝑀 (𝑛+ ) = 𝐸 [𝐷 (𝑛+ )] = 𝐸 [𝐷 (𝑛+ ) ; 𝑏 > 𝑛+ ] + 𝐸 [𝐷 (𝑛+ ) ; 𝑏 ≀ 𝑛+ ]

Multiplying (22) by 𝑒𝑛 and summing over 𝑛 result in

π‘š0 (𝑒) =

1 βˆ’ [πœƒV (πœ†π‘’)]

𝐽

1 βˆ’ πœƒV (πœ†π‘’) ∞ π‘š

π‘šβˆ’π‘™ π‘š β‹… βˆ‘ βˆ‘π‘ƒ {𝑉 = π‘š} π‘’π‘š ( ) πœ†π‘™ πœ† π‘šπ‘™ (𝑒) 𝑙 π‘š=1

= π‘Šπ‘– (𝑛+ ) + 𝐸 [𝐷 (𝑏 ) ; 𝑏 ≀ 𝑛+ ]

𝑙=1

𝑛

+ π‘š1 (𝑒) β‹…

+ βˆ‘π‘ƒ {𝑏 = π‘˜} 𝐸 [𝐷 ((𝑛 βˆ’ π‘˜)+ )]

πœ†π‘’ (1 βˆ’ πœƒ) V (πœ†π‘’) 1 βˆ’ πœ†π‘’

π‘˜=𝑖 𝑛

β‹…

= π‘Šπ‘– (𝑛+ ) + 𝑇𝑖 (𝑛+ ) + βˆ‘π‘ƒ {𝑏 = π‘˜} 𝑀 ((𝑛 βˆ’ π‘˜)+ ) . π‘˜=𝑖

(20) Taking 𝑧-transform on both sides of the above equation and applying Lemma 4 give 𝑑𝑖 (𝑒) + 𝑀𝑖 (𝑒) =

[1 βˆ’ 𝐡𝑖 (𝑒)] 𝐺 (𝑒) (1 βˆ’ 𝑒) [1 βˆ’ 𝐺 (𝑒)]

.

(21)

For 𝑀0 (𝑛+ ), by using the same method used for the solution of π‘Žπ‘– (𝑒) (𝑖 β‰₯ 0), it gets 𝑀0 (𝑛+ ) = 𝐸 [𝐷 (𝑛+ ) | 𝑁 (0+ ) = 0] = 𝐸 [𝐷 (𝑛+ ) ; 𝜏 ≀ 𝑛+ ] 𝐽

= βˆ‘πΈ [𝐷 (𝑛+ ) ; 𝜏 ≀ 𝑛+ ; 𝑉 < 𝜏 ≀ 𝑉 ]

β‹…

1 βˆ’ [πœƒV (πœ†π‘’)]

(23)

π½βˆ’1

+ π‘š1 (𝑒)

1 βˆ’ πœƒV (πœ†π‘’) πœ†π‘’ β‹… πœƒπ½βˆ’1 [V (πœ†π‘’)] 1 βˆ’ πœ†π‘’

𝐽

.

For 𝑀𝑖 (𝑛+ ) (𝑖 β‰₯ 1), it has 𝑀𝑖 (𝑛+ ) = 𝐸 [𝐷 (𝑛+ ) | 𝑁 (0+ ) = 𝑖] = 𝐸 [𝐷 (𝑛+ ) ; 𝑛+ < 𝑏 ] + 𝐸 [𝐷 (𝑛+ ) ; 𝑏 ≀ 𝑛+ ] = π‘Šπ‘– (𝑛+ ) + 𝐸 {The number of departures during

𝑖=1

(0+ , 𝑏 ] ; 𝑏 ≀ 𝑛+ }

+ 𝐸 [𝐷 (𝑛+ ) ; 𝜏 ≀ 𝑛+ ; 𝑉 < 𝜏] 𝐽

= βˆ‘πœƒπ‘–βˆ’1 𝐸 [𝐷 (𝑛+ ) ; 𝑉 ≀ 𝑛+ ; 𝑉 < 𝜏 ≀ 𝑉 ] 𝑖=1

+ 𝐸 {The number of departures during (𝑏 , 𝑛+ ] ; 𝑏 ≀ 𝑛+ }

𝐽

+ βˆ‘πœƒπ‘–βˆ’2 (1 βˆ’ πœƒ) 𝐸 [𝐷 (𝑛+ ) ; 𝜏 ≀ 𝑛+ ; 𝑉 < 𝜏] 𝑖=2

𝑛

= π‘Šπ‘– (𝑛+ ) + 𝑇𝑖 (𝑛+ ) + βˆ‘π‘ƒ {𝑏 = π‘˜} 𝑀0 ((𝑛 βˆ’ π‘˜)+ ) . π‘˜=𝑖

𝑛

+ πœƒπ½βˆ’1 βˆ‘π‘ƒ {𝜏 = 𝑖} 𝑃 {𝑉 < 𝑖} β‹… 𝑀1 ((𝑛 βˆ’ 𝑖)+ )

(24)

𝑖=1

𝐽

∞

𝑛

= βˆ‘πœƒπ‘–βˆ’1 βˆ‘ βˆ‘ 𝑃 {𝑉 = π‘˜}

Taking 𝑧-transform on both sides of (24) and applying (21) yield

π‘š=1π‘˜=π‘–βˆ’1

𝑖=1 π‘›βˆ’π‘˜

π‘™βˆ’π‘š 𝑙 Γ— βˆ‘ 𝑃 {𝑉 = 𝑙} β‹… 𝑃 {𝜏 > π‘˜} ( ) πœ†π‘š πœ† π‘š 𝑙=1

+

𝐽

π‘–βˆ’2

β‹… π‘€π‘š ((𝑛 βˆ’ π‘˜ βˆ’ 𝑙) ) + βˆ‘πœƒ

π‘šπ‘– (𝑒) = (1 βˆ’ πœƒ)

[1 βˆ’ 𝐡𝑖 (𝑒)] 𝐺 (𝑒) (1 βˆ’ 𝑒) [1 βˆ’ 𝐺 (𝑒)]

+ 𝐡𝑖 (𝑒) π‘š0 (𝑒) .

(25)

𝑖=2

𝑛

Solving the simultaneous equations (23) and (25), it gets the expression of π‘šπ‘– (𝑒) (𝑖 β‰₯ 0). From the expressions of π‘šπ‘– (𝑒) (𝑖 β‰₯ 0) given here, π‘Žπ‘– (𝑒) (𝑖 β‰₯ 0) given by Theorem 2, and π‘š(𝑒) given by Lemma 4, the relation among π‘šπ‘– (𝑒), π‘š(𝑒), and π‘Žπ‘– (𝑒) is established as follows:

Γ— βˆ‘ 𝑃 {𝜏 = π‘˜} 𝑃 {𝑉 < π‘˜} π‘˜=1

β‹… 𝑀1 ((𝑛 βˆ’ π‘˜)+ ) + πœƒπ½βˆ’1 𝑛

Γ— βˆ‘π‘ƒ {𝜏 = 𝑖} 𝑃 {𝑉 < 𝑖} β‹… 𝑀1 ((𝑛 βˆ’ 𝑖)+ ) . 𝑖=1

(22)

π‘šπ‘– (𝑒) = π‘š (𝑒) β‹… (1 βˆ’ 𝑒) π‘Žπ‘– (𝑒) .

(26)

6

Discrete Dynamics in Nature and Society

Applying Theorem 2 and Lemma 4, when condition 𝜌 < 1 holds, it has the following steady result: 𝑀𝑖 (𝑛+ ) = limβˆ’ (1 βˆ’ 𝑒)2 π‘šπ‘– (𝑒) π‘›β†’βˆž 𝑒→1 𝑛 lim

= limβˆ’ (1 βˆ’ 𝑒)2 π‘š (𝑒) β‹… limβˆ’ (1 βˆ’ 𝑒) π‘Žπ‘– (𝑒) 𝑒→1

𝑒→1

+

)+1 πœ’π‘– ; thus 𝑆𝐷(𝑛+ )+1 denotes the Proof. Let 𝑆𝐷(𝑛+ )+1 = βˆ‘π·(𝑛 𝑖=1 epoch immediately after the (𝐷(𝑛+ )+1)th service completion. + Denote by 𝑋(𝑛 ) the remain service time of a customer being served at epoch 𝑛+ with the corresponding steady state 𝑋+ = + lim 𝑋(𝑛 ) and the steady state distribution π‘₯𝑖 = 𝑃{𝑋+ = π‘›β†’βˆž

𝑖 𝑖}, 𝑖 β‰₯ 1, which has P.G.F. π‘‹βˆ— (𝑒) = βˆ‘βˆž 𝑖=1 π‘₯𝑖 𝑒 . From [25], it has

𝑀 (𝑛+ ) β‹… lim 𝐴 (𝑛+ ) π‘›β†’βˆž π‘›β†’βˆž 𝑛

= lim

1 { β‹… 𝜌 = πœ†, { {𝛼 ={ {1 { β‹… 1, {𝛼

π‘‹βˆ— (𝑒) =

𝜌