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1 queue with N-policy
Applied Mathematical Modelling 37 (2013) 4643–4652
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Analysis of the GI/Geo/1 queue with N-policy Dae-Eun Lim a, Doo Ho Lee b,⇑, Won Seok Yang c, Kyung-Chul Chae d a
Division of Business and Commerce, Baekseok University, Chungnam 330–704, South Korea Big Data Software Research Laboratory, ETRI, Daejeon, 305–700, South Korea Department of Business Administration, Hannam University, Daejeon 306–791, South Korea d Department of Industrial and Systems Engineering, KAIST, Daejeon, 305–701, South Korea b c
a r t i c l e
i n f o
Article history: Received 20 March 2012 Received in revised form 8 August 2012 Accepted 11 September 2012 Available online 27 September 2012 Keywords: Discrete-time queue General input queue N-policy Busy period
a b s t r a c t We consider a discrete-time single server N-policy GI=Geo=1 queueing system. The server stops servicing whenever the system becomes empty, and resumes its service as soon as the number of waiting customers in the queue reaches N. Using an embedded Markov chain and a trial solution approach, the stationary queue length distribution at arrival epochs is obtained. Furthermore, we obtain the stationary queue length distribution at arbitrary epochs by using the preceding result and a semi-Markov process. The sojourn time distribution is also presented. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction After its introduction in the work of Yandin and Naor [1], N-policy has been studied extensively. Most studies concerning N-policy have focused on queueing systems with continuous-time queueing systems (See the references in Moreno [2] for M=G=1 queues and Ke [3], Ke and Wang [4], Ke et al. [5] and Zhang and Tian [6] for GI=M=1 queues.). Recently, N-policy has been extended to the discrete-time Geo=G=1 queues (the discrete counterpart of the M=G=1 queue) by Moreno [2,7] and Hernández-Diáz and Moreno [8]. Also, discrete-time GI=Geo=1 queues (the discrete counterpart of the GI=M=1 queue) with various vacation policies such as multiple vacations [9], a single vacation [10] and working vacations [11–13] have been vigorously investigated. Refer to Alfa [14] for the references of the vacation models in discrete-time queueing systems. However, N-policy has been studied only by Lee and Chae [15] and their analysis was incorrect. First, they confused an assumption regarding the order in which arrivals and departures take place, namely the late arrival system with delayed access (LAS-DA; henceforth we refer to LAS-DA as LAS) and the early arrival system (EAS) (about the EAS and LAS, refer to Hunter [16]). Second, their regenerative approach to obtain a trial solution is not correct at the step of making relations between probabilities at arrival epochs and at arbitrary epochs. In this paper, we analyse one of the less investigated queueing system models: the GI=Geo=1 queue with N-policy. This system operates as follows: customers arrive according to a renewal process. The server stops servicing whenever the system becomes empty and resumes its service as soon as the queue length reaches N. We rectify the two flaws in Lee and Chae [15] and thoroughly discuss the differences between GI=Geo=1 with N-policy using the LAS and EAS. Analysing discrete-time queueing systems with general input is significant for the following reasons. If the volatility of an arrival process is considered to be significant, these models can be used to explore the effect of the volatility. On the other hand, discrete-time queueing systems have wide applicability in computer and digital communication systems (refer to ⇑ Corresponding author. Tel.: +82 42 860 3887. E-mail address: [email protected] (D.H. Lee). 0307-904X/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2012.09.037
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Bruneel and Kim [17] and references therein). In discrete-time queues, time is assumed to be divided into intervals of the same length called slots. At this point, it is said that discrete-time queues are more appropriate to depict digital communication systems due to the packetized nature of transport protocols. Consider a wireless sensor network (WSN) that consists of a number of distributed sensor nodes and a sink node. A sensor can communicate with the other sensors in a possible communication range via wireless links. Most sensor nodes are equipped with non-rechargeable batteries that have limited lifetime. Since the lifetime of a WSN is dominated by sensor nodes, one of the important design issues is managing power consumption of these batteries for prolonging the lifetime. All generated packets are eventually delivered to the sink node for final processing. Sensor nodes located nearer to the sink node deplete their energy faster since they carry heavier traffic loads heading for the sink node. Heavy traffic results in frequent switching its mode from idle (or busy) to busy (or idle). Consequently, it is important to decrease the number of changes to extend the lifetime of sensor nodes. The N-policy is very helpful to alleviate the number of changes, even though it increases the average queue length. For more details, we encourage readers to refer to Jiang et al. [18]. The manufacturing system is another major application of the N-policy. This policy has been applied to remanufacturing system [19], due date setting in a make-to-order production environment [20,21], preventive maintenance of a machine [22] and so on. The remainder of this paper is organised as follows. In Section 2, assumptions and the mathematical model are introduced. Section 3 analyses the model and presents queue length distributions at arbitrary and arrival epochs. The waiting time distribution is also presented in Section 3. Section 4 introduces the method that determines the optimal N value to minimize the operating cost per unit time. Numerical results are followed in Section 5. Finally, Section 6 concludes the article. 2. Preliminaries Throughout this paper, we primarily adopt the EAS system. However, we also make comparisons to the LAS system which always appears as Remarks. Let the time axis be marked by k ¼ 0; 1; . . .. According to the EAS, a potential arrival takes place þ þ in ðk; k Þ and a potential departure occurs in ðk ; kÞ where k ¼ 1; 2; . . .. A service is assumed to start only at k ; k ¼ 0; 1; . . ., þ and to end at k ; k ¼ 1; 2; . . .. Assuming that a service starts at k and that the length of this service is lðP 1Þ, then the service will end at m where m ¼ k þ l. Interarrival times fAn ; n P 1g are independent and identically distributed (i.i.d.) general discrete random variables. A distribution, a mean and its probability generating function (PGF) is denoted by P i PfAn ¼ ig ¼ ai ði P 1Þ; E½An ¼ 1=k, and AðzÞ ¼ 1 i¼1 ai z , respectively. The service times fSn ; n P 1g are also i.i.d. random variables and follow a geometric distribution with parameter l:
PfSn ¼ ig ¼ si ¼ li1 l;
i ¼ 1; 2; . . . ;
0 < l < 1; P1
l ¼ 1 l:
zÞ. Service times and interarrival times are mutually indeIts PGF is denoted by SðzÞ and is defined by i¼1 si z ¼ lz=ð1 l pendent and k < l is assumed for the system to be stable. The GI=Geo=1 with the N-policy operates as follows. Suppose that a customer leaves the system in ðk ; kÞ; k ¼ 1; 2; . . ., and that this departure leaves the system empty. Then, the server waits until the number of waiting customers reaches N. As soon as the number of waiting customer reaches N, the server recommences servicing customers. i
þ
Remark 1. For the LAS system, we assume that a potential arrival and departure occurs in ðk ; kÞ and ðk; k Þ, respectively. A service can start and end only at k; k ¼ 0; 1; 2; . . ., and m; m ¼ 1; 2; . . ., respectively, where the length of the service is lðP 1Þ and m ¼ k þ l. h 3. Stationary distributions We now proceed to obtain three important distributions: the queue length distributions at arrival and arbitrary epochs and the sojourn time distribution. 3.1. Embedded Markov chain Let Ln denote the number of customers in the system at the nth arrival instant and define
Jn ¼
0; the server is not available at the n-th arrival instant; 1; the server is available at the n-th arrival instant:
Then, fðLn ; J n Þ; n P 1g is a Markov chain. Let a random variable Bn denote the number of customers served during An . Its distribution function is defined as follows:
PfBn ¼ kg ¼ bk ¼
1 X P½Bn ¼ kjAn ¼ iP½An ¼ i ¼ i¼1
1 X i¼maxð1;kÞ
ai
i k
lk l ik :
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0
In addition, bk is defined by: 0
bk ¼ 1
k1 X
bi :
i¼0
In order to express the transition probability matrix of fðLn ; J n Þ; n P 1g, we define
Pði;jÞ;ðk;lÞ ¼ PfLnþ1 ¼ k; J nþ1 ¼ ljLn ¼ i; J n ¼ jg;
j; l 2 f0; 1g;
0 6 k 6 i þ 1;
0 6 i 6 N 1 when j ¼ 0 and i P 1 when j ¼ 1: Here, the transition probabilities are considered. Transition probabilities can be classified by the availability of a server before and after of a transition. First, consider transitions when the server is not available. Insomuch as the server is not available, only an arrival can occur that is transitions from ði; 0Þ to ði þ 1; 0Þ where 0 6 i 6 N 2 occur with probability 1. Thus, we have
Pði;0Þ;ðiþ1;0Þ ¼ 1;
0 6 i 6 N 2:
The second case is that the state of the server is not available before and becomes available after a transition, specifically, þ from ðN 1; 0Þ to ði; 1Þ, where 1 6 i 6 N. Suppose that the nth arrival occurs in ðk; k Þ and that the state ðLn ; J n Þ is equal to ðN 1; 0Þ. Moreover, the very next arrival is assumed to occur after AðP 1Þ slots. The nth arrival causes the server to start þ servicing at k , then the maximum number of departures is equal to A because the first departure can occur in ððk þ 1Þ ; k þ 1Þ. Therefore, we have
PðN1;0Þ;ði;1Þ ¼ bNi ;
1 6 i 6 N:
Remark 2. Under the LAS assumption, transition probabilities for this case should be reconsidered. Let the example be repeated except that a customer arrived in ðk ; kÞ. Since the length of service time is assumed to be greater than or equal to 1 þ slot, the first departure can not occur in ðk; k Þ. Thus,the maximum number of departures is A 1, not A. Here, we define dk and dk 0 as follows:
dk ¼
1 X i1 ai lk l i1k ; k i¼kþ1
0
dk ¼ 1
k1 X
di :
i¼0
Then, under the LAS assumption, we have
PðN1;0Þ;ði;1Þ ¼ dNi ;
1 6 i 6 N:
The third case includes transitions when the server is available. Transitions from ði; 1Þ to ði þ 1 n; 1Þ occur if there are n service completions during an interarrival time where 0 6 n 6 i. Thus, we have
Pði;1Þ;ðiþ1n;1Þ ¼ bn ;
0 6 n 6 i:
The last case corresponds to transitions from ði; jÞ to (0,0), where ði; jÞ 2 fði; jÞji P j; j ¼ 1g [ fðN 1; 0Þg. These probabilities can be found easily since each row sum of a transition probability matrix should be 1. Thus, these probabilities are expressed 0 by corresponding biþ1 ði P 2Þ. For example, consider P ð2;1Þ;ð0;0Þ . If a customer arrived at the system that was in ð2; 1Þ, then possible states for the very next customer are ð1; 1Þ; ð2; 1Þ; ð3; 1Þ and (0,0). Probabilities P ð2;1Þ;ð1;1Þ ; Pð2;1Þ;ð2;1Þ , and Pð2;1Þ;ð3;1Þ are 0 previously defined by b2 ; b1 and b0 , respectively. Therefore, P ð2;1Þ;ð0;0Þ ¼ 1 ðb0 þ b1 þ b2 Þ ¼ b3 . 3.2. Stationary queue length distributions at arrival epochs Let pi;j denote the stationary probability that an arriving customer finds i customers in the system that is in state j where ði; jÞ 2 fði; jÞji P j; j ¼ 0; 1g n fði; 0Þji P Ng. We define
P ¼ ðp0;0 ; p1;0 ; . . . ; pN1;0 ; p1;1 ; p2;1 ; . . . ; pN;1 ; . . .Þ: These probabilities should satisfy the following two equations
P ¼ PP; N1 X
1 X
i¼0
i¼1
pi;0 þ
pi;1 ¼ 1;
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where P is the transition probability matrix of the Markov chain. Since one of the balance equations is redundant, we ignore a column corresponding to the state (0, 0). Then, the rest are as follows:
p0;0 ¼ p1;0 ¼ p2;0 ¼ ¼ pN1;0 ; p1;1 ¼ pN1;0 bN1 þ pn;1 ¼ pN1;0 bNn þ
1 X i¼1 1 X
ð1aÞ
pi;1 bi ;
ð1bÞ
piþn1;1 bi ; 2 6 n 6 N;
ð1cÞ
i¼0
pn;1 ¼
1 X
piþn1;1 bi ; n P N þ 1:
ð1dÞ
i¼0
Remark 3. Under the LAS assumption, the Eqs. (1b) and (1c) are respectively replaced by
p1;1 ¼ pN1;0 dN1 þ pn;1 ¼ pN1;0 dNn þ
1 X
pi;1 bi ;
ð2aÞ
i¼1 1 X
piþn1;1 bi ; 2 6 n 6 N:
ð2bÞ
i¼0
(1a) is self-evident and now we proceed to consider (1b)–(1d). We use the trial solution as follows:
pn;1 ¼ p0;0 C N rn ; n P N;
ð3Þ
where C N will be determined later. Since we are assuming stable system, r is the unique root within jrj < 1 of the equation below [16],
r¼
1 X
ri bi ¼ Aðl þ lrÞ:
ð4Þ
i¼0
Note that we can also obtain (4) by substituting pn;1 s ðn P NÞ of (3) into (1d). Next, (1b) and (1c) can be solved successively starting from where n ¼ N to n ¼ 1. For instance, consider n ¼ N in (1c), we have
pN;1 ¼ pN1;0 b0 þ pN1;1 b0 þ
1 X
piþN1;1 bi :
i¼1
According to (1a) and (3), the equations Applying three equations above gives
pN;1 ¼ p0;0 C N rN ; pN1;0 ¼ p0;0 and piþN1;1 ¼ p0;0 C N riþN1 are obtained, respectively.
p0;0 C N rN ¼ p0;0 b0 þ pN1;1 b0 þ p0;0 C N rN1 ðr b0 Þ: Additionally, the equation
pN1;1 ¼ p0;0 ðC N r The rest,
N1
P1
i¼1 r
i
bi ¼
P1
i¼0 r
i
bi b0 ¼ r b0 is also used. Therefore,
1Þ:
pn;1 ð1 6 n 6 N 2Þ, can be obtained in the same manner by defining C n ðn P 1Þ as follows:
C 1 ¼ 1; b0 C 2 ¼ C 1 ; b0 C n ¼ C n1
n1 X bnk C k ;
n P 3:
k¼2
Alternatively, for n P 2,
C n ¼ C 2 bn1 þ C 3 bn2 þ þ C nþ1 b0 :
ð5Þ
The stationary queue length distribution at arrival epochs is summarized in the following theorem. Theorem 3.1
pn;0 ¼ p0;0 ; 1 6 n 6 N 1; p ðC rn C Nn Þ; 1 6 n 6 N 1; pn;1 ¼ 0;0 N n p0;0 C N r ; n P N; where C 1 ¼ 1; b0 C 2 ¼ C 1 and b0 C n ¼ C n1
Pn1
k¼2 bnk C k ;
n P 3.
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p0;0 can be obtained by the normalization condition:
p0;0 ¼
1r ! : N1 X C n þ rC N ð1 rÞ N n¼1
Remark 4. Considering the LAS, after the same procedures as the EAS case, the probabilities at arrival epochs are obtained as follows:
(
p
LAS n;1
¼
n pLAS 1 6 n 6 N 1; 0;0 ðK N r K Nn Þ; n pLAS K r ; n P N; 0;0 N
where each pLAS i;j has the same meaning as the corresponding C n ðn P 1Þ is replaced by K n ðn P 1Þ and those are defined as
pi;j of the EAS. The probabilities are in the same form except
b0 K 1 ¼ d0 ; b0 K n ¼ K n1
n1 X bnk K k þ dn1 ;
n P 2:
k¼1
Consequently,
pLAS 0;0 ¼
pLAS 0;0 is changed as follows:
1r ! ; N1 X ð1 rÞ N K n þ rK N
ð6Þ
n¼1
where di ði P 0Þ is defined in Remark 2. h 3.3. Stationary queue length distributions at arbitrary epochs Let Q ðtÞ and L denote the system size at time tðt P 0Þ and at arbitrary epochs. In discrete-time queues, the system size þ þ does not change during ðk ; ðk þ 1Þ Þ; k ¼ 0; 1; 2; . . .. Thus, we regard k points of time as arbitrary epochs. þ
PfL ¼ ng ¼ lim PfQ ðk Þ ¼ ng: k!1
We define a new process fðZðtÞ; KðtÞÞ; t P 0g where ZðtÞ denotes the system size immediately after the most recent arrival and KðtÞ equals 0 (or 1) if the most recent arrival sees the server is unavailable (or available). Recall that we assumed that a þ þ potential departure and arrival occurs in ðk ; kÞ and ðk; k Þ, respectively. Zðk Þ is the number of customers after all possible þ events. Suppose that the most recent arrival is the n-th arrival and that it occured in ðk; k Þ. We define Q n as Q n ¼ Q ðkÞ. In other words, Q n denotes the system size observed by the nth customer. Thus, þ
þ
Q n þ 1 ¼ Qðk Þ ¼ Zðk Þ: fðZðtÞ; KðtÞÞ; t P 0g is a semi-Markov process (SMP) having fðQ n þ 1; J n Þ; n P 1g for its embedded Markov chain. þ AE is defined as the elapsed interarrival time at arbitrary epochs, i.e., at k when k ! 1, in steady-state. The probability mass function of AE is known as PfAE ¼ ig ¼ kPfA > ig where A is the interarrival time and i ¼ 0; 1; . . .. Let D be the number of departures during AE and di ¼ PfD ¼ ig. Additionally, P i;j is defined as the probability that the system is in ði; jÞ state at arbitrary epochs where ði; jÞ 2 fði; jÞji P j; j ¼ 0; 1g n fði; 0Þji P Ng. Theorem 3.2. The probability distribution at arbitrary epochs is given by:
P0;0 ¼ 1 q ðN 1Þp0;0 ; Pnþ1;0 ¼ p0;0 ;
where q ¼ k=l
0 6 n 6 N 2;
lPnþ1;1 ¼ kðpn;1 þ pn;0 Þ; 0 6 n 6 N 1; lPnþ1;1 ¼ kpn;1 ; n P N:
ð7aÞ ð7bÞ ð7cÞ ð7dÞ
Proof. In order to derive P i;j , we consider three cases depending upon the system state and the relationship between fQ ðtÞ; t P 0g and fðZðtÞ; KðtÞÞ; t P 0g. First, (7b) can be explained by the renewal reward theorem [23]. Let C denote the length of a cycle from the instant the number of customers in the system becomes 1 to the instant the system becomes empty again for the first time (more detailed discussion of a cycle is given in Section 4). C n0 is defined as the time duration that the system is in the state ðn; 0Þ
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during a cycle. Then, it is clear that E½C n0 ¼ 1=kð0 6 n 6 N 1Þ. An additional random variable K is defined as the number of customers that arrive during a cycle C. During a cycle, only one customer sees the state ð0; 0Þ on arrival, thus the number average probability p0;0 can be written in terms of E½K as follows:
p0;0 ¼
1 : E½K
Since customers are never blocked, the number of customers arrived during a cycle kE½C is equal to the number of customers served during the cycle E½K. Finally, we have
p0;0 ¼
1 1 ¼ E½K kE½C
Similar argument can be found in Medhi [24]. By the renewal reward theorem [23],
Pn;0 ¼
E½C n0 ; E½C
n P 1:
ð8Þ
Substituting E½C n0 ¼ 1=k and 1=E½C ¼ kp0;0 into (8) results in (7b). Finally, we proceed to obtain P 0;0 . For single server queues, the following relation holds [25].
E½fraction of busy servers ¼ 1 Pfthe server is busyg ¼ q: Using the above probability and (7b), the following equation gives (7a).
P0;0 þ
N1 X Pi;0 ¼ Pfthe server is idleg: i¼1
Subsequently, we consider cases when the system is available and these are subdivided depending upon whether the system size is above N. We first derive P nþ1;1 for n P N, the easier part. Using the relationship between fQ ðtÞ; t P 0g and fðZðtÞ; KðtÞÞ; t P 0g, we have
Pnþ1;1 ¼
1 X
1 X
1 X din rin
i¼n
i¼n
i¼n
pi;1 din ¼
p0;0 C N ri din ¼ p0;0 C N rn
The left-hand side is the probability that the system size is ðn þ 1Þ at an arbitrary epoch. The right-hand side is equal to the probability that the system size is i at an arrival epoch immediately before the arbitrary epoch. ði nÞ customers are served and leave the system during the time duration from this arrival epoch to the arbitrary epoch. Note that the time duration is denoted by AE . Here, we define two terms as follows:
DðzÞ ¼
1 X d i zi ; i¼0
aðzÞ ¼
1 X
PfAE ¼ igzi :
i¼0
aðzÞ is known as f1 AðzÞg=fð1 zÞE½Ag and DðzÞ is easily found since it is the PGF of Bernoulli events during AE [26]. Therefore, we have
þ lzÞ ¼ DðzÞ ¼ aðl
þ lzÞ 1 Að l : þ lzÞgE½A f1 ðl
Substituting r into DðzÞ yields q and finally we have (7d) using DðrÞ ¼ q. We now consider the final case where the system is available with less than N customers. Using the same argument as Pnþ1;1 for n P N gives
Pn;1 ¼ pN1;0 dNn þ
1 X
pi;1 diðn1Þ ; 1 6 n 6 N:
ð9Þ
i¼maxð1;n1Þ
Solving (9) results in (7c) (a detailed proof is in the Appendix).
h
Remark 5. The Eqs. (7c) and (7d) can be intuitively clear. The left-hand sides of (7c) and (7d) denote the expected decrease of customers from n þ 1 to n per unit time. On the other hand, the right-hand sides denote the expected increase from n to n þ 1 per unit time. In a stable system, these two expected values should be equal. h
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Remark 6. With the same argument as Remark 5, the stationary queue length distribution at arbitrary epochs under the LAS assumption is as follows: LAS PLAS 0;0 ¼ 1 q ðN 1Þp0;0 ;
PLAS n;0
LAS n1;0 ;
¼p
l
PLAS 1;1
l
PLAS nþ1;1
¼k
lPLAS nþ1;1
where q ¼ k=l;
1 6 n 6 N 1; þ pLAS 0;0 ;
LAS 1;1
lp
LAS LAS ¼ k lpLAS 1 6 n 6 N 1; nþ1;1 þ lpn;1 þ pn;0 ; LAS ¼ k lpLAS n P N: nþ1;1 þ lpn;1 ;
Complexities in this distribution stem from the peculiar assumption of the LAS system that a departure occurs after an arrival. h Remark 7. Stationary queue length distributions at arbitrary and at arrival epochs of 1-policy queue coincide with the standard GI=Geo=1 queue (for both the EAS and the LAS). h 3.4. Stationary sojourn time distribution In this section, we derive the stationary sojourn time distribution under the first-come first-served (FCFS) discipline where the sojourn time is defined as the sum of the queue waiting time and the service time. Let W and WðzÞ denote the sojourn time and its PGF, respectively. Then, the sojourn time distribution is presented in the following theorem. Theorem 3.3
WðzÞ ¼ p0;0 SðzÞ where
p1 ðzÞ ¼
P1
n¼1
AðzÞN SðzÞN þ p1 ðSðzÞÞ SðzÞ; AðzÞ SðzÞ
ð10Þ
pn;1 zn and SðzÞ ¼ lz=ð1 l zÞ.
Proof. Assume that an arriving customer sees the server idle and nð0 6 n 6 N 1Þ customers, then the customer must wait for other N ðn þ 1Þ customers until the queue length becomes N. After the system starts its service, the customer must wait in the queue for the service times of n customers ahead. For the sojourn time, the customer’s own service time is added to the waiting time. On the other hand, suppose an arriving customer finds the system busy and n customers. Then, the sojourn time equals the service times of n þ 1 customers (i.e., other n customers and the customer’s own service time). Combining all these things, we have
WðzÞ ¼
N1 X
1 X
n¼0
n¼1
pn;0 fAðzÞgNn1 fSðzÞgnþ1 þ
pn;1 fSðzÞgnþ1 :
ð11Þ
Note that (11) is identical to (10). h Corollary 1. The expected waiting time is given by:
dWð1Þ p0;0 ¼ E½W ¼ dz l
(
) N1 X NðN 1Þ ðl þ kÞ N þ CN þ ðn þ 1ÞC Nn : 2k ð1 rÞ2 n¼1 rð2 rÞ
Corollary 2. Little’s law can be verified by defining and deriving two PGFs as follows:
P0 ðzÞ ¼
N1 X z zN Pi;0 zi ¼ 1 q ðN 1Þp0;0 þ p0;0 ; 1z i¼0
! 1 N1 X X 1 zN rz i n Pi;1 z ¼ qzp0;0 C Nn z ; þ CN P1 ðzÞ ¼ 1 rz n¼1 1z i¼1 PðzÞ ¼ P0 ðzÞ þ P1 ðzÞ:
ð12Þ
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Remark 8. Under the LAS assumption, the sojourn time is almost the same as (10), but the second term is slightly changed. Suppose that a customer arrives between ðk ; kÞ and sees n customers. Then, a customer who is being served at that time can þ depart between ðk; k Þ. Thus, the residual service time can be 0 at an arrival epoch and its PGF is not SðzÞ but z1 SðzÞ. Therefore, the sojourn time of the GI=Geo=1 with the LAS is as follows:
WðzÞ ¼ pð0;0Þ
N1 1 X X N1n nþ1 þ pðn;1Þ fSðzÞgnþ1 z1 : fAðzÞg fSðzÞg n¼0
n¼1
Also, it is easy to verify that Little’s law also holds under the LAS assumption. h Remark 9. The sojourn time distribution of 1-policy queue with the EAS (or the LAS) coincides with the corresponding standard GI=Geo=1 queue with the EAS (or the LAS) assumption. Note that the expected sojourn time of GI=Geo=1 is unique regardless of whether the assumption is the EAS or the LAS. h
4. Optimal control of the N-policy In this section, the optimal control of N-policy queueing system is demonstrated. To review the optimal design and control of queues with various policies such as N; T and D, refer to Tadj and Choudhury [27]. We define a cycle and the cost structure first. A cycle consists of a busy and a consecutive idle period. A busy period is the time duration from the instant when the number of customers reaches N to the instant the system becomes empty for the first time. An idle period followed by a busy period is the time duration from the instant when a busy period ends until the number of customers in the system reaches N. We define a regenerative cycle by considering the starting point of a busy period as a regeneration point. Note that the definition of a cycle is consistent in Section 3.3. We consider the total cost per unit time and it is assumed to be the sum of two cost terms: setup and holding cost. The setup cost occurs once when a busy period starts and the holding cost occurs per unit time and per customer during a busy period. Then, the objective function for minimizing the total cost per unit time is as follows:
2 minTC ¼ min NP1
6 6 Cs Cs þ C h L ¼ min6 NP1 6 E½C 4
NP1
kð1 rÞ k 1r ! ! þ Ch N1 N1 X X l ð1 rÞ N ð1 rÞ N C n þ rC N C n þ rC N n¼1
3 ( ) N1 7 X rð2 rÞ NðN 1Þ 7 þ ðn þ 1ÞC Nn 7 N þ CN ðl þ kÞ 2 5 2k ð1 rÞ n¼1
n¼1
ð13Þ
where C; L; C s and C h denote the length of a cycle, the average number of customers, setup and holding cost, respectively. The N is subject to a positive integer. L can be easily found from (12) and Little’s law.
Remark 10. Consider that the system size becomes 0 by a departure in ðk ; kÞ under the EAS assumption. Suppose that a þ customer arrives in ðk; k Þ, immediately after the departure, and that the customer finds the system empty. In such a case, for mathematical tractability, we have assumed that a busy period ends at k, and that the arriving customer must wait until the number of customer reaches N. However, under the previous assumption, the length of a busy period can be considerably different from the LAS result. We can modify the previous assumption as follows. Under the same situation, let the service of an arriving customer in þ þ ðk; k Þ start from k . It means that the customer does not wait for other N 1 customers but generates a new busy period after the busy period that ended at k. Here, these two (or more) consecutive busy periods are regarded as one busy period. þ When a busy period ends at k, a new busy period starts at k according to geometric distribution with parameter k since a þ customer arrives at the system in ðk; k Þ with probability k. Also, (8) should be modified. h 5. Numerical results In this section, numerical results of the GI=Geo=1 queues with the above cost structure are presented. To examine the effect of the volatility of the arrival process, two distributions with the same mean of 5 but different coefficient of variations (CVs) are used. They are geometric distributions (CV = 0.89) and the negative binomial distributions (CV = 0.37) with q ¼ 0:5 or q ¼ 0:9, respectively. We assumed that setup cost C s and holding cost C h as 1 and 0.01, respectively. Fig. 1 shows that the distribution with a larger CV has a lower optimal N value under the same traffic intensity.
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0.7 Geo (rho=0.5) NB (rho=0.5) Geo (rho=0.9) 0.6
NB (rho=0.9)
Total Cost
0.5
0.4
0.3
0.2
0.1 0
2
4
6
8
10
N - policy Fig. 1. Total costs over traffic intensity (q) and N policy.
6. Conclusion We have analysed a discrete-time single server GI=Geo=1 queueing system with N policy and presented important performance measures such as stationary queue length distributions and sojourn time distribution. Especially, the stationary queue length distributions at arbitrary epochs of general input queues with the N-policy have been uncommonly presented in most studies regardless of continuous- or discrete-time queueing systems. The queue length distribution at arbitrary epochs of the continuous counterpart GI=M=1 queue can be derived from our result using the methods presented in Chae et al. [11] or Moreno [2]. Furthermore, differences between the EAS and LAS results were also discussed. In addition, a method to obtain an optimal N value that minimizes the total cost under a certain cost structure was presented. Finally, numerical results that examine the effect of the volatility of the arrival process were given. Appendix A. The proof of (7c) First, we use the relation
di ¼
P½B > i ¼ qP½B > i; E½B
where B ¼ limn!1 Bn . Recall that di denotes a probability that the number of departures during AE is i. Then, from the equation above the followings are easily obtained.
qbi ¼
q d0
;
i ¼ 0;
di1 di
;
i P 1:
ð14Þ
According to (5), C n ð1 6 n 6 N 1Þ can be written as follows:
C Nn ¼ C Nnþ1 b0 þ C Nn b1 þ þ C 2 bNn1 :
ð15Þ
Multiplying q on both sides of (15) and using (14) gives
qC Nn ¼ qC Nðn1Þ
N2 X i¼n1
C Ni diðn1Þ þ
N2 X C Ni din ; i¼n
1 6 n 6 N 1:
ð16Þ
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Further, the following relation can be derived recursively starting from when n ¼ N 1. N2 X
C Ni diðn1Þ ¼ q C Nðn1Þ C 1 ;
2 6 n 6 N 1:
ð17Þ
i¼n1
Applying Theorem 3.1 and DðzÞjz¼r ¼ q into (9) yields
(
Pn;1 ¼ p0;0 dNn þ C N rn1
1 X i¼n1
r iðn1Þ diðn1Þ
N1 X
) C Ni diðn1Þ
i¼n1
( ¼ p0;0 qC N rn1
N2 X
) C Ni diðn1Þ :
ð18Þ
i¼n1
Substituting (17) into (18) and using Theorem 3.1 results in (7c), which completes the proof. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
M. Yadin, P. Naor, Queueing systems with a removable service station, Oper. Res. Q. 14 (1963) 393–405. P. Moreno, A discrete-time single-server queue with a modified N-policy, Int. J. Syst. Sci. 38 (2007) 483–492. J.C. Ke, The analysis of a general input queue with N policy and exponential vacations, Queueing. Syst. 45 (2003) 135–160. J.C. Ke, K.H. Wang, A recursive method for the N policy GI/M/1 queueing system with finite capacity, Eur. J. Oper. Res. 142 (2002) 577–594. J.C. Ke, K.H. Wang, C.H. Liou, A single vacation model G/M/1/K with N threshold policy, Sankhya Ser. A 68 (2006) 198–226. Z.G. Zhang, N. Tian, The N threshold policy for the GI/M/1 queue, Oper. Res. Lett. 32 (2004) 77–84. P. Moreno, Analysis of a Geo/G/1 queueing system with a generalized N-policy and setup-closedown times, QTQM 5 (2008) 111128. A.G. Hernández-Diáz, P. Moreno, A discrete-time single-server queueing system with an N-policy, an early setup and a generalization of the Bernoulli feedback, Math. Comput. Model. 49 (2009) 977–990. N. Tian, Z.G. Zhang, The discrete-time GI/Geo/1 queue with multiple vacations, Queueing. Syst. 40 (2002) 283–294. K.C. Chae, S.M. Lee, S.H. Lee, The discrete-time GI/Geo/1 queue with single geometric vacation, QTQM 5 (2008) 21–31. K.C. Chae, D.E. Lim, W.S. Yang, The GI/M/1 queue and the GI/Geo/1 queue both with single working vacation, Perform. Eval. 66 (2009) 356–367. J.H. Li, N.S. Tian, The discrete-time GI/Geo/1 queue with working vacations and vacation interruption, Appl. Math. Comput. 185 (2007) 1–10. J.H. Li, N.S. Tian, W.Y. Liu, Discrete-time GI/Geo/1 queue with multiple working vacations, Queueing Syst. 56 (2007) 53–63. A.S. Alfa, Vacation models in discrete time, Queueing Syst. 44 (2003) 5–30. S.M. Lee, K.C. Chae, Analysis of the GI/Geom/1 system with N-policy, in: Proceedings of the Fifth Asia-Pacific Industrial Engineering and Management Systems Conference, Goldcoast, Australia, 2004, pp. 32.6.1–32.6.6. . J.J. Hunter, Mathematical techniques of applied probability, Discrete-time Models: Techniques and Applications, 2, Academic Press, New York, 1983. H. Bruneel, B.G. Kim, Discrete-Time Models for Communication Systems Including ATM, Kluwer Academic Publishers, Norwell, MA, 1992. F.C. Jiang, D.C. Huang, C.T. Yang, F.Y. Leu, Lifetime elongation for wireless sensor network using queue-based approaches, J. Supercomput. 59 (2011) 1– 24. H.K. Aksoy, S.M. Gupta, Near optimal buffer allocation in remanufacturing systems with N-policy, Comput. Ind. Eng. 59 (2010) 496–508. N.P. Dellaert, Multi-item production control for production to order, Eng. Cost. Prod. Econ. 17 (1989) 167–173. N.P. Dellaert, Due-date setting and production control, Int. J. Prod. Econ. 23 (1991) 59–67. S.M. Gupta, N-policy queueing system with finite population, Trans. Oper. Res. 7 (1995) 45–62. R.W. Wolff, Stochastic Modeling and the Theory of Queues, Prentice-Hall, New York, 1989. J. Medhi, Stochastic Models in Queueing Theory, Academic Press, 2002. Second Ed. L. KleinRock, Queueing Systems, Theory, John Wiley & Sons, 1975. vol. 1. H. Takagi, Queueing Analysis, North-Holland, 1993. vol. 3. L. Tadj, G. Choudhury, Optimal design and control of queues, Top 13 (2005) 359–412.
Contents lists available at SciVerse ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Analysis of the GI/Geo/1 queue with N-policy Dae-Eun Lim a, Doo Ho Lee b,⇑, Won Seok Yang c, Kyung-Chul Chae d a
Division of Business and Commerce, Baekseok University, Chungnam 330–704, South Korea Big Data Software Research Laboratory, ETRI, Daejeon, 305–700, South Korea Department of Business Administration, Hannam University, Daejeon 306–791, South Korea d Department of Industrial and Systems Engineering, KAIST, Daejeon, 305–701, South Korea b c
a r t i c l e
i n f o
Article history: Received 20 March 2012 Received in revised form 8 August 2012 Accepted 11 September 2012 Available online 27 September 2012 Keywords: Discrete-time queue General input queue N-policy Busy period
a b s t r a c t We consider a discrete-time single server N-policy GI=Geo=1 queueing system. The server stops servicing whenever the system becomes empty, and resumes its service as soon as the number of waiting customers in the queue reaches N. Using an embedded Markov chain and a trial solution approach, the stationary queue length distribution at arrival epochs is obtained. Furthermore, we obtain the stationary queue length distribution at arbitrary epochs by using the preceding result and a semi-Markov process. The sojourn time distribution is also presented. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction After its introduction in the work of Yandin and Naor [1], N-policy has been studied extensively. Most studies concerning N-policy have focused on queueing systems with continuous-time queueing systems (See the references in Moreno [2] for M=G=1 queues and Ke [3], Ke and Wang [4], Ke et al. [5] and Zhang and Tian [6] for GI=M=1 queues.). Recently, N-policy has been extended to the discrete-time Geo=G=1 queues (the discrete counterpart of the M=G=1 queue) by Moreno [2,7] and Hernández-Diáz and Moreno [8]. Also, discrete-time GI=Geo=1 queues (the discrete counterpart of the GI=M=1 queue) with various vacation policies such as multiple vacations [9], a single vacation [10] and working vacations [11–13] have been vigorously investigated. Refer to Alfa [14] for the references of the vacation models in discrete-time queueing systems. However, N-policy has been studied only by Lee and Chae [15] and their analysis was incorrect. First, they confused an assumption regarding the order in which arrivals and departures take place, namely the late arrival system with delayed access (LAS-DA; henceforth we refer to LAS-DA as LAS) and the early arrival system (EAS) (about the EAS and LAS, refer to Hunter [16]). Second, their regenerative approach to obtain a trial solution is not correct at the step of making relations between probabilities at arrival epochs and at arbitrary epochs. In this paper, we analyse one of the less investigated queueing system models: the GI=Geo=1 queue with N-policy. This system operates as follows: customers arrive according to a renewal process. The server stops servicing whenever the system becomes empty and resumes its service as soon as the queue length reaches N. We rectify the two flaws in Lee and Chae [15] and thoroughly discuss the differences between GI=Geo=1 with N-policy using the LAS and EAS. Analysing discrete-time queueing systems with general input is significant for the following reasons. If the volatility of an arrival process is considered to be significant, these models can be used to explore the effect of the volatility. On the other hand, discrete-time queueing systems have wide applicability in computer and digital communication systems (refer to ⇑ Corresponding author. Tel.: +82 42 860 3887. E-mail address: [email protected] (D.H. Lee). 0307-904X/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2012.09.037
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Bruneel and Kim [17] and references therein). In discrete-time queues, time is assumed to be divided into intervals of the same length called slots. At this point, it is said that discrete-time queues are more appropriate to depict digital communication systems due to the packetized nature of transport protocols. Consider a wireless sensor network (WSN) that consists of a number of distributed sensor nodes and a sink node. A sensor can communicate with the other sensors in a possible communication range via wireless links. Most sensor nodes are equipped with non-rechargeable batteries that have limited lifetime. Since the lifetime of a WSN is dominated by sensor nodes, one of the important design issues is managing power consumption of these batteries for prolonging the lifetime. All generated packets are eventually delivered to the sink node for final processing. Sensor nodes located nearer to the sink node deplete their energy faster since they carry heavier traffic loads heading for the sink node. Heavy traffic results in frequent switching its mode from idle (or busy) to busy (or idle). Consequently, it is important to decrease the number of changes to extend the lifetime of sensor nodes. The N-policy is very helpful to alleviate the number of changes, even though it increases the average queue length. For more details, we encourage readers to refer to Jiang et al. [18]. The manufacturing system is another major application of the N-policy. This policy has been applied to remanufacturing system [19], due date setting in a make-to-order production environment [20,21], preventive maintenance of a machine [22] and so on. The remainder of this paper is organised as follows. In Section 2, assumptions and the mathematical model are introduced. Section 3 analyses the model and presents queue length distributions at arbitrary and arrival epochs. The waiting time distribution is also presented in Section 3. Section 4 introduces the method that determines the optimal N value to minimize the operating cost per unit time. Numerical results are followed in Section 5. Finally, Section 6 concludes the article. 2. Preliminaries Throughout this paper, we primarily adopt the EAS system. However, we also make comparisons to the LAS system which always appears as Remarks. Let the time axis be marked by k ¼ 0; 1; . . .. According to the EAS, a potential arrival takes place þ þ in ðk; k Þ and a potential departure occurs in ðk ; kÞ where k ¼ 1; 2; . . .. A service is assumed to start only at k ; k ¼ 0; 1; . . ., þ and to end at k ; k ¼ 1; 2; . . .. Assuming that a service starts at k and that the length of this service is lðP 1Þ, then the service will end at m where m ¼ k þ l. Interarrival times fAn ; n P 1g are independent and identically distributed (i.i.d.) general discrete random variables. A distribution, a mean and its probability generating function (PGF) is denoted by P i PfAn ¼ ig ¼ ai ði P 1Þ; E½An ¼ 1=k, and AðzÞ ¼ 1 i¼1 ai z , respectively. The service times fSn ; n P 1g are also i.i.d. random variables and follow a geometric distribution with parameter l:
PfSn ¼ ig ¼ si ¼ li1 l;
i ¼ 1; 2; . . . ;
0 < l < 1; P1
l ¼ 1 l:
zÞ. Service times and interarrival times are mutually indeIts PGF is denoted by SðzÞ and is defined by i¼1 si z ¼ lz=ð1 l pendent and k < l is assumed for the system to be stable. The GI=Geo=1 with the N-policy operates as follows. Suppose that a customer leaves the system in ðk ; kÞ; k ¼ 1; 2; . . ., and that this departure leaves the system empty. Then, the server waits until the number of waiting customers reaches N. As soon as the number of waiting customer reaches N, the server recommences servicing customers. i
þ
Remark 1. For the LAS system, we assume that a potential arrival and departure occurs in ðk ; kÞ and ðk; k Þ, respectively. A service can start and end only at k; k ¼ 0; 1; 2; . . ., and m; m ¼ 1; 2; . . ., respectively, where the length of the service is lðP 1Þ and m ¼ k þ l. h 3. Stationary distributions We now proceed to obtain three important distributions: the queue length distributions at arrival and arbitrary epochs and the sojourn time distribution. 3.1. Embedded Markov chain Let Ln denote the number of customers in the system at the nth arrival instant and define
Jn ¼
0; the server is not available at the n-th arrival instant; 1; the server is available at the n-th arrival instant:
Then, fðLn ; J n Þ; n P 1g is a Markov chain. Let a random variable Bn denote the number of customers served during An . Its distribution function is defined as follows:
PfBn ¼ kg ¼ bk ¼
1 X P½Bn ¼ kjAn ¼ iP½An ¼ i ¼ i¼1
1 X i¼maxð1;kÞ
ai
i k
lk l ik :
D.-E. Lim et al. / Applied Mathematical Modelling 37 (2013) 4643–4652
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0
In addition, bk is defined by: 0
bk ¼ 1
k1 X
bi :
i¼0
In order to express the transition probability matrix of fðLn ; J n Þ; n P 1g, we define
Pði;jÞ;ðk;lÞ ¼ PfLnþ1 ¼ k; J nþ1 ¼ ljLn ¼ i; J n ¼ jg;
j; l 2 f0; 1g;
0 6 k 6 i þ 1;
0 6 i 6 N 1 when j ¼ 0 and i P 1 when j ¼ 1: Here, the transition probabilities are considered. Transition probabilities can be classified by the availability of a server before and after of a transition. First, consider transitions when the server is not available. Insomuch as the server is not available, only an arrival can occur that is transitions from ði; 0Þ to ði þ 1; 0Þ where 0 6 i 6 N 2 occur with probability 1. Thus, we have
Pði;0Þ;ðiþ1;0Þ ¼ 1;
0 6 i 6 N 2:
The second case is that the state of the server is not available before and becomes available after a transition, specifically, þ from ðN 1; 0Þ to ði; 1Þ, where 1 6 i 6 N. Suppose that the nth arrival occurs in ðk; k Þ and that the state ðLn ; J n Þ is equal to ðN 1; 0Þ. Moreover, the very next arrival is assumed to occur after AðP 1Þ slots. The nth arrival causes the server to start þ servicing at k , then the maximum number of departures is equal to A because the first departure can occur in ððk þ 1Þ ; k þ 1Þ. Therefore, we have
PðN1;0Þ;ði;1Þ ¼ bNi ;
1 6 i 6 N:
Remark 2. Under the LAS assumption, transition probabilities for this case should be reconsidered. Let the example be repeated except that a customer arrived in ðk ; kÞ. Since the length of service time is assumed to be greater than or equal to 1 þ slot, the first departure can not occur in ðk; k Þ. Thus,the maximum number of departures is A 1, not A. Here, we define dk and dk 0 as follows:
dk ¼
1 X i1 ai lk l i1k ; k i¼kþ1
0
dk ¼ 1
k1 X
di :
i¼0
Then, under the LAS assumption, we have
PðN1;0Þ;ði;1Þ ¼ dNi ;
1 6 i 6 N:
The third case includes transitions when the server is available. Transitions from ði; 1Þ to ði þ 1 n; 1Þ occur if there are n service completions during an interarrival time where 0 6 n 6 i. Thus, we have
Pði;1Þ;ðiþ1n;1Þ ¼ bn ;
0 6 n 6 i:
The last case corresponds to transitions from ði; jÞ to (0,0), where ði; jÞ 2 fði; jÞji P j; j ¼ 1g [ fðN 1; 0Þg. These probabilities can be found easily since each row sum of a transition probability matrix should be 1. Thus, these probabilities are expressed 0 by corresponding biþ1 ði P 2Þ. For example, consider P ð2;1Þ;ð0;0Þ . If a customer arrived at the system that was in ð2; 1Þ, then possible states for the very next customer are ð1; 1Þ; ð2; 1Þ; ð3; 1Þ and (0,0). Probabilities P ð2;1Þ;ð1;1Þ ; Pð2;1Þ;ð2;1Þ , and Pð2;1Þ;ð3;1Þ are 0 previously defined by b2 ; b1 and b0 , respectively. Therefore, P ð2;1Þ;ð0;0Þ ¼ 1 ðb0 þ b1 þ b2 Þ ¼ b3 . 3.2. Stationary queue length distributions at arrival epochs Let pi;j denote the stationary probability that an arriving customer finds i customers in the system that is in state j where ði; jÞ 2 fði; jÞji P j; j ¼ 0; 1g n fði; 0Þji P Ng. We define
P ¼ ðp0;0 ; p1;0 ; . . . ; pN1;0 ; p1;1 ; p2;1 ; . . . ; pN;1 ; . . .Þ: These probabilities should satisfy the following two equations
P ¼ PP; N1 X
1 X
i¼0
i¼1
pi;0 þ
pi;1 ¼ 1;
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where P is the transition probability matrix of the Markov chain. Since one of the balance equations is redundant, we ignore a column corresponding to the state (0, 0). Then, the rest are as follows:
p0;0 ¼ p1;0 ¼ p2;0 ¼ ¼ pN1;0 ; p1;1 ¼ pN1;0 bN1 þ pn;1 ¼ pN1;0 bNn þ
1 X i¼1 1 X
ð1aÞ
pi;1 bi ;
ð1bÞ
piþn1;1 bi ; 2 6 n 6 N;
ð1cÞ
i¼0
pn;1 ¼
1 X
piþn1;1 bi ; n P N þ 1:
ð1dÞ
i¼0
Remark 3. Under the LAS assumption, the Eqs. (1b) and (1c) are respectively replaced by
p1;1 ¼ pN1;0 dN1 þ pn;1 ¼ pN1;0 dNn þ
1 X
pi;1 bi ;
ð2aÞ
i¼1 1 X
piþn1;1 bi ; 2 6 n 6 N:
ð2bÞ
i¼0
(1a) is self-evident and now we proceed to consider (1b)–(1d). We use the trial solution as follows:
pn;1 ¼ p0;0 C N rn ; n P N;
ð3Þ
where C N will be determined later. Since we are assuming stable system, r is the unique root within jrj < 1 of the equation below [16],
r¼
1 X
ri bi ¼ Aðl þ lrÞ:
ð4Þ
i¼0
Note that we can also obtain (4) by substituting pn;1 s ðn P NÞ of (3) into (1d). Next, (1b) and (1c) can be solved successively starting from where n ¼ N to n ¼ 1. For instance, consider n ¼ N in (1c), we have
pN;1 ¼ pN1;0 b0 þ pN1;1 b0 þ
1 X
piþN1;1 bi :
i¼1
According to (1a) and (3), the equations Applying three equations above gives
pN;1 ¼ p0;0 C N rN ; pN1;0 ¼ p0;0 and piþN1;1 ¼ p0;0 C N riþN1 are obtained, respectively.
p0;0 C N rN ¼ p0;0 b0 þ pN1;1 b0 þ p0;0 C N rN1 ðr b0 Þ: Additionally, the equation
pN1;1 ¼ p0;0 ðC N r The rest,
N1
P1
i¼1 r
i
bi ¼
P1
i¼0 r
i
bi b0 ¼ r b0 is also used. Therefore,
1Þ:
pn;1 ð1 6 n 6 N 2Þ, can be obtained in the same manner by defining C n ðn P 1Þ as follows:
C 1 ¼ 1; b0 C 2 ¼ C 1 ; b0 C n ¼ C n1
n1 X bnk C k ;
n P 3:
k¼2
Alternatively, for n P 2,
C n ¼ C 2 bn1 þ C 3 bn2 þ þ C nþ1 b0 :
ð5Þ
The stationary queue length distribution at arrival epochs is summarized in the following theorem. Theorem 3.1
pn;0 ¼ p0;0 ; 1 6 n 6 N 1; p ðC rn C Nn Þ; 1 6 n 6 N 1; pn;1 ¼ 0;0 N n p0;0 C N r ; n P N; where C 1 ¼ 1; b0 C 2 ¼ C 1 and b0 C n ¼ C n1
Pn1
k¼2 bnk C k ;
n P 3.
D.-E. Lim et al. / Applied Mathematical Modelling 37 (2013) 4643–4652
4647
p0;0 can be obtained by the normalization condition:
p0;0 ¼
1r ! : N1 X C n þ rC N ð1 rÞ N n¼1
Remark 4. Considering the LAS, after the same procedures as the EAS case, the probabilities at arrival epochs are obtained as follows:
(
p
LAS n;1
¼
n pLAS 1 6 n 6 N 1; 0;0 ðK N r K Nn Þ; n pLAS K r ; n P N; 0;0 N
where each pLAS i;j has the same meaning as the corresponding C n ðn P 1Þ is replaced by K n ðn P 1Þ and those are defined as
pi;j of the EAS. The probabilities are in the same form except
b0 K 1 ¼ d0 ; b0 K n ¼ K n1
n1 X bnk K k þ dn1 ;
n P 2:
k¼1
Consequently,
pLAS 0;0 ¼
pLAS 0;0 is changed as follows:
1r ! ; N1 X ð1 rÞ N K n þ rK N
ð6Þ
n¼1
where di ði P 0Þ is defined in Remark 2. h 3.3. Stationary queue length distributions at arbitrary epochs Let Q ðtÞ and L denote the system size at time tðt P 0Þ and at arbitrary epochs. In discrete-time queues, the system size þ þ does not change during ðk ; ðk þ 1Þ Þ; k ¼ 0; 1; 2; . . .. Thus, we regard k points of time as arbitrary epochs. þ
PfL ¼ ng ¼ lim PfQ ðk Þ ¼ ng: k!1
We define a new process fðZðtÞ; KðtÞÞ; t P 0g where ZðtÞ denotes the system size immediately after the most recent arrival and KðtÞ equals 0 (or 1) if the most recent arrival sees the server is unavailable (or available). Recall that we assumed that a þ þ potential departure and arrival occurs in ðk ; kÞ and ðk; k Þ, respectively. Zðk Þ is the number of customers after all possible þ events. Suppose that the most recent arrival is the n-th arrival and that it occured in ðk; k Þ. We define Q n as Q n ¼ Q ðkÞ. In other words, Q n denotes the system size observed by the nth customer. Thus, þ
þ
Q n þ 1 ¼ Qðk Þ ¼ Zðk Þ: fðZðtÞ; KðtÞÞ; t P 0g is a semi-Markov process (SMP) having fðQ n þ 1; J n Þ; n P 1g for its embedded Markov chain. þ AE is defined as the elapsed interarrival time at arbitrary epochs, i.e., at k when k ! 1, in steady-state. The probability mass function of AE is known as PfAE ¼ ig ¼ kPfA > ig where A is the interarrival time and i ¼ 0; 1; . . .. Let D be the number of departures during AE and di ¼ PfD ¼ ig. Additionally, P i;j is defined as the probability that the system is in ði; jÞ state at arbitrary epochs where ði; jÞ 2 fði; jÞji P j; j ¼ 0; 1g n fði; 0Þji P Ng. Theorem 3.2. The probability distribution at arbitrary epochs is given by:
P0;0 ¼ 1 q ðN 1Þp0;0 ; Pnþ1;0 ¼ p0;0 ;
where q ¼ k=l
0 6 n 6 N 2;
lPnþ1;1 ¼ kðpn;1 þ pn;0 Þ; 0 6 n 6 N 1; lPnþ1;1 ¼ kpn;1 ; n P N:
ð7aÞ ð7bÞ ð7cÞ ð7dÞ
Proof. In order to derive P i;j , we consider three cases depending upon the system state and the relationship between fQ ðtÞ; t P 0g and fðZðtÞ; KðtÞÞ; t P 0g. First, (7b) can be explained by the renewal reward theorem [23]. Let C denote the length of a cycle from the instant the number of customers in the system becomes 1 to the instant the system becomes empty again for the first time (more detailed discussion of a cycle is given in Section 4). C n0 is defined as the time duration that the system is in the state ðn; 0Þ
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during a cycle. Then, it is clear that E½C n0 ¼ 1=kð0 6 n 6 N 1Þ. An additional random variable K is defined as the number of customers that arrive during a cycle C. During a cycle, only one customer sees the state ð0; 0Þ on arrival, thus the number average probability p0;0 can be written in terms of E½K as follows:
p0;0 ¼
1 : E½K
Since customers are never blocked, the number of customers arrived during a cycle kE½C is equal to the number of customers served during the cycle E½K. Finally, we have
p0;0 ¼
1 1 ¼ E½K kE½C
Similar argument can be found in Medhi [24]. By the renewal reward theorem [23],
Pn;0 ¼
E½C n0 ; E½C
n P 1:
ð8Þ
Substituting E½C n0 ¼ 1=k and 1=E½C ¼ kp0;0 into (8) results in (7b). Finally, we proceed to obtain P 0;0 . For single server queues, the following relation holds [25].
E½fraction of busy servers ¼ 1 Pfthe server is busyg ¼ q: Using the above probability and (7b), the following equation gives (7a).
P0;0 þ
N1 X Pi;0 ¼ Pfthe server is idleg: i¼1
Subsequently, we consider cases when the system is available and these are subdivided depending upon whether the system size is above N. We first derive P nþ1;1 for n P N, the easier part. Using the relationship between fQ ðtÞ; t P 0g and fðZðtÞ; KðtÞÞ; t P 0g, we have
Pnþ1;1 ¼
1 X
1 X
1 X din rin
i¼n
i¼n
i¼n
pi;1 din ¼
p0;0 C N ri din ¼ p0;0 C N rn
The left-hand side is the probability that the system size is ðn þ 1Þ at an arbitrary epoch. The right-hand side is equal to the probability that the system size is i at an arrival epoch immediately before the arbitrary epoch. ði nÞ customers are served and leave the system during the time duration from this arrival epoch to the arbitrary epoch. Note that the time duration is denoted by AE . Here, we define two terms as follows:
DðzÞ ¼
1 X d i zi ; i¼0
aðzÞ ¼
1 X
PfAE ¼ igzi :
i¼0
aðzÞ is known as f1 AðzÞg=fð1 zÞE½Ag and DðzÞ is easily found since it is the PGF of Bernoulli events during AE [26]. Therefore, we have
þ lzÞ ¼ DðzÞ ¼ aðl
þ lzÞ 1 Að l : þ lzÞgE½A f1 ðl
Substituting r into DðzÞ yields q and finally we have (7d) using DðrÞ ¼ q. We now consider the final case where the system is available with less than N customers. Using the same argument as Pnþ1;1 for n P N gives
Pn;1 ¼ pN1;0 dNn þ
1 X
pi;1 diðn1Þ ; 1 6 n 6 N:
ð9Þ
i¼maxð1;n1Þ
Solving (9) results in (7c) (a detailed proof is in the Appendix).
h
Remark 5. The Eqs. (7c) and (7d) can be intuitively clear. The left-hand sides of (7c) and (7d) denote the expected decrease of customers from n þ 1 to n per unit time. On the other hand, the right-hand sides denote the expected increase from n to n þ 1 per unit time. In a stable system, these two expected values should be equal. h
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Remark 6. With the same argument as Remark 5, the stationary queue length distribution at arbitrary epochs under the LAS assumption is as follows: LAS PLAS 0;0 ¼ 1 q ðN 1Þp0;0 ;
PLAS n;0
LAS n1;0 ;
¼p
l
PLAS 1;1
l
PLAS nþ1;1
¼k
lPLAS nþ1;1
where q ¼ k=l;
1 6 n 6 N 1; þ pLAS 0;0 ;
LAS 1;1
lp
LAS LAS ¼ k lpLAS 1 6 n 6 N 1; nþ1;1 þ lpn;1 þ pn;0 ; LAS ¼ k lpLAS n P N: nþ1;1 þ lpn;1 ;
Complexities in this distribution stem from the peculiar assumption of the LAS system that a departure occurs after an arrival. h Remark 7. Stationary queue length distributions at arbitrary and at arrival epochs of 1-policy queue coincide with the standard GI=Geo=1 queue (for both the EAS and the LAS). h 3.4. Stationary sojourn time distribution In this section, we derive the stationary sojourn time distribution under the first-come first-served (FCFS) discipline where the sojourn time is defined as the sum of the queue waiting time and the service time. Let W and WðzÞ denote the sojourn time and its PGF, respectively. Then, the sojourn time distribution is presented in the following theorem. Theorem 3.3
WðzÞ ¼ p0;0 SðzÞ where
p1 ðzÞ ¼
P1
n¼1
AðzÞN SðzÞN þ p1 ðSðzÞÞ SðzÞ; AðzÞ SðzÞ
ð10Þ
pn;1 zn and SðzÞ ¼ lz=ð1 l zÞ.
Proof. Assume that an arriving customer sees the server idle and nð0 6 n 6 N 1Þ customers, then the customer must wait for other N ðn þ 1Þ customers until the queue length becomes N. After the system starts its service, the customer must wait in the queue for the service times of n customers ahead. For the sojourn time, the customer’s own service time is added to the waiting time. On the other hand, suppose an arriving customer finds the system busy and n customers. Then, the sojourn time equals the service times of n þ 1 customers (i.e., other n customers and the customer’s own service time). Combining all these things, we have
WðzÞ ¼
N1 X
1 X
n¼0
n¼1
pn;0 fAðzÞgNn1 fSðzÞgnþ1 þ
pn;1 fSðzÞgnþ1 :
ð11Þ
Note that (11) is identical to (10). h Corollary 1. The expected waiting time is given by:
dWð1Þ p0;0 ¼ E½W ¼ dz l
(
) N1 X NðN 1Þ ðl þ kÞ N þ CN þ ðn þ 1ÞC Nn : 2k ð1 rÞ2 n¼1 rð2 rÞ
Corollary 2. Little’s law can be verified by defining and deriving two PGFs as follows:
P0 ðzÞ ¼
N1 X z zN Pi;0 zi ¼ 1 q ðN 1Þp0;0 þ p0;0 ; 1z i¼0
! 1 N1 X X 1 zN rz i n Pi;1 z ¼ qzp0;0 C Nn z ; þ CN P1 ðzÞ ¼ 1 rz n¼1 1z i¼1 PðzÞ ¼ P0 ðzÞ þ P1 ðzÞ:
ð12Þ
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Remark 8. Under the LAS assumption, the sojourn time is almost the same as (10), but the second term is slightly changed. Suppose that a customer arrives between ðk ; kÞ and sees n customers. Then, a customer who is being served at that time can þ depart between ðk; k Þ. Thus, the residual service time can be 0 at an arrival epoch and its PGF is not SðzÞ but z1 SðzÞ. Therefore, the sojourn time of the GI=Geo=1 with the LAS is as follows:
WðzÞ ¼ pð0;0Þ
N1 1 X X N1n nþ1 þ pðn;1Þ fSðzÞgnþ1 z1 : fAðzÞg fSðzÞg n¼0
n¼1
Also, it is easy to verify that Little’s law also holds under the LAS assumption. h Remark 9. The sojourn time distribution of 1-policy queue with the EAS (or the LAS) coincides with the corresponding standard GI=Geo=1 queue with the EAS (or the LAS) assumption. Note that the expected sojourn time of GI=Geo=1 is unique regardless of whether the assumption is the EAS or the LAS. h
4. Optimal control of the N-policy In this section, the optimal control of N-policy queueing system is demonstrated. To review the optimal design and control of queues with various policies such as N; T and D, refer to Tadj and Choudhury [27]. We define a cycle and the cost structure first. A cycle consists of a busy and a consecutive idle period. A busy period is the time duration from the instant when the number of customers reaches N to the instant the system becomes empty for the first time. An idle period followed by a busy period is the time duration from the instant when a busy period ends until the number of customers in the system reaches N. We define a regenerative cycle by considering the starting point of a busy period as a regeneration point. Note that the definition of a cycle is consistent in Section 3.3. We consider the total cost per unit time and it is assumed to be the sum of two cost terms: setup and holding cost. The setup cost occurs once when a busy period starts and the holding cost occurs per unit time and per customer during a busy period. Then, the objective function for minimizing the total cost per unit time is as follows:
2 minTC ¼ min NP1
6 6 Cs Cs þ C h L ¼ min6 NP1 6 E½C 4
NP1
kð1 rÞ k 1r ! ! þ Ch N1 N1 X X l ð1 rÞ N ð1 rÞ N C n þ rC N C n þ rC N n¼1
3 ( ) N1 7 X rð2 rÞ NðN 1Þ 7 þ ðn þ 1ÞC Nn 7 N þ CN ðl þ kÞ 2 5 2k ð1 rÞ n¼1
n¼1
ð13Þ
where C; L; C s and C h denote the length of a cycle, the average number of customers, setup and holding cost, respectively. The N is subject to a positive integer. L can be easily found from (12) and Little’s law.
Remark 10. Consider that the system size becomes 0 by a departure in ðk ; kÞ under the EAS assumption. Suppose that a þ customer arrives in ðk; k Þ, immediately after the departure, and that the customer finds the system empty. In such a case, for mathematical tractability, we have assumed that a busy period ends at k, and that the arriving customer must wait until the number of customer reaches N. However, under the previous assumption, the length of a busy period can be considerably different from the LAS result. We can modify the previous assumption as follows. Under the same situation, let the service of an arriving customer in þ þ ðk; k Þ start from k . It means that the customer does not wait for other N 1 customers but generates a new busy period after the busy period that ended at k. Here, these two (or more) consecutive busy periods are regarded as one busy period. þ When a busy period ends at k, a new busy period starts at k according to geometric distribution with parameter k since a þ customer arrives at the system in ðk; k Þ with probability k. Also, (8) should be modified. h 5. Numerical results In this section, numerical results of the GI=Geo=1 queues with the above cost structure are presented. To examine the effect of the volatility of the arrival process, two distributions with the same mean of 5 but different coefficient of variations (CVs) are used. They are geometric distributions (CV = 0.89) and the negative binomial distributions (CV = 0.37) with q ¼ 0:5 or q ¼ 0:9, respectively. We assumed that setup cost C s and holding cost C h as 1 and 0.01, respectively. Fig. 1 shows that the distribution with a larger CV has a lower optimal N value under the same traffic intensity.
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0.7 Geo (rho=0.5) NB (rho=0.5) Geo (rho=0.9) 0.6
NB (rho=0.9)
Total Cost
0.5
0.4
0.3
0.2
0.1 0
2
4
6
8
10
N - policy Fig. 1. Total costs over traffic intensity (q) and N policy.
6. Conclusion We have analysed a discrete-time single server GI=Geo=1 queueing system with N policy and presented important performance measures such as stationary queue length distributions and sojourn time distribution. Especially, the stationary queue length distributions at arbitrary epochs of general input queues with the N-policy have been uncommonly presented in most studies regardless of continuous- or discrete-time queueing systems. The queue length distribution at arbitrary epochs of the continuous counterpart GI=M=1 queue can be derived from our result using the methods presented in Chae et al. [11] or Moreno [2]. Furthermore, differences between the EAS and LAS results were also discussed. In addition, a method to obtain an optimal N value that minimizes the total cost under a certain cost structure was presented. Finally, numerical results that examine the effect of the volatility of the arrival process were given. Appendix A. The proof of (7c) First, we use the relation
di ¼
P½B > i ¼ qP½B > i; E½B
where B ¼ limn!1 Bn . Recall that di denotes a probability that the number of departures during AE is i. Then, from the equation above the followings are easily obtained.
qbi ¼
q d0
;
i ¼ 0;
di1 di
;
i P 1:
ð14Þ
According to (5), C n ð1 6 n 6 N 1Þ can be written as follows:
C Nn ¼ C Nnþ1 b0 þ C Nn b1 þ þ C 2 bNn1 :
ð15Þ
Multiplying q on both sides of (15) and using (14) gives
qC Nn ¼ qC Nðn1Þ
N2 X i¼n1
C Ni diðn1Þ þ
N2 X C Ni din ; i¼n
1 6 n 6 N 1:
ð16Þ
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Further, the following relation can be derived recursively starting from when n ¼ N 1. N2 X
C Ni diðn1Þ ¼ q C Nðn1Þ C 1 ;
2 6 n 6 N 1:
ð17Þ
i¼n1
Applying Theorem 3.1 and DðzÞjz¼r ¼ q into (9) yields
(
Pn;1 ¼ p0;0 dNn þ C N rn1
1 X i¼n1
r iðn1Þ diðn1Þ
N1 X
) C Ni diðn1Þ
i¼n1
( ¼ p0;0 qC N rn1
N2 X
) C Ni diðn1Þ :
ð18Þ
i¼n1
Substituting (17) into (18) and using Theorem 3.1 results in (7c), which completes the proof. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
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