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1 retrial queueing system with priority services
European Journal of Operational Research 256 (2017) 178–186
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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Stochastics and Statistics
A Geo/G/1 retrial queueing system with priority services I. Atencia University of Málaga, Higher Polytechnic School, Department of Applied Mathematics, 29071 Málaga, Spain
a r t i c l e
i n f o
Article history: Received 30 January 2016 Accepted 8 July 2016 Available online 14 July 2016 Keywords: Discrete-time Retrials Approximation to continuous-time Service upgrade Sojourn times
a b s t r a c t This paper considers a discrete-time retrial queueing system in which the arriving customers can decide to go directly to the server expelling out of the system the customer that is currently being served, if any, or to join the orbit in accordance with a FCFS discipline. An extensive analysis of the model has been carried out, and using a generating functions approach, the distributions of the number of customers in the orbit and in the system with its respective means are obtained. The stochastic decomposition law has been derived, and, as an application, bounds for the proximity between the steady-state distributions for the considered queueing system and its corresponding standard system are obtained. Also, recursive formulae for calculating the steady-state distributions of the orbit and system size have been developed. Besides, we prove that the M/G/1 retrial queue with service interruptions can be approximated by the corresponding discrete-time system. The generating function of the sojourn time of a customer in the orbit and in the system have also been provided. Finally, some numerical examples to illustrate the effect of the parameters on several performance characteristics and a section of conclusions commenting the main research contributions of this paper are presented. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Queueing systems that are characterized by the fact that a customer finding all the servers occupied upon arrival must abandon the service area, but some time later it returns to re-initiate its demand, are called retrial queueing systems. This type of customers repeat their attempts for service from a pool of unsatisfied customers (called orbit or retrial group) and are superimposed on the normal stream of external arrivals. Apart from theoretical interests, retrial queues have been successfully applied to telephone switching systems, telecommunication networks and computer systems: packet switch networks, shared bus local area networks operating under the Carrier-Sense Multiple Access protocol and collision avoidance star local area networks. A review of the main results, literature and discussion of practical situations where retrial queues arise can be located in the books by Falin and Templeton (1997) and Artalejo and Gomez-Corral (2008). At first, the study of the queues with repeated attempts was focused on the continuous-time, but Yang and Li (1995) extended the analysis to discrete-time systems giving insight to many researchers, for example, Choi and Kim (1997), Atencia and Moreno (2004), Moreno and Nobel (2008) and Wu, Wang, and Liu (2013). It is probably fair to say that the difficulty of the research on retrial models is partly due to their intractability, because from a practical point of
E-mail address: [email protected] http://dx.doi.org/10.1016/j.ejor.2016.07.011 0377-2217/© 2016 Elsevier B.V. All rights reserved.
view retrial models often describe a more realistic picture of many queueing situations than other types of models. Another feature that is usually found when a message is being processed in computers, in communications switching queues, etc., is that sometimes the information incoming to the server is more actual that the one on service. In that case, the message is moved to another place if the contained information can be used later on or if the information is not any more valuable it is deleted, in both cases the server is interrupted and upgraded. For a general survey on queues with service interruptions one can refer the papers Fiems, Steyaert, and Bruneel (2002), Walraevens, Steyaert, and Bruneel (2006), Krishnamoorthy, Gopakumar, and Viswanath Narayanan (2012), Krishnamoorthy, Pramod, and Chakravarthy (2013) and Atencia (2015), and for a detailed review on queues with service interruptions refer to the paper Krishnamoorthy, Pramod, and Chakravarthy (2014). The mechanism of moving messages by the arrival of one of them is called synchronized or triggered motion. There are several mechanisms on how and where the messages are moved, for a survey on them refer to Artalejo (20 0 0), Gelenbe and Label (1998) and Atencia (2014). In some manufacturing processes, the time that a part may spend in a buffer between successive operations is limited. Therefore, an important topic in delay models is the study of the sojourn-time distribution of an individual job. The probability distribution, rather than just the mean, or just the mean and
I. Atencia / European Journal of Operational Research 256 (2017) 178–186
standard deviation, is important because a factory’s profitability requires that no more than a specified percentage of the parts are scrapped. This issue is important in several industries, including the semiconductor, food, chemical, and steel industries, see for example, Shi and Gershwin (2016). The remainder of this paper is structured as follows. The assumptions of the queueing system under study are given in the next section. In Section 3, we study the Markov chain. The queue and system size distribution are obtained together with several performance measures of the system. In Section 4, a stochastic decomposition property, and, as an application, upper and lower estimates for the distance between the steady-state distribution for the considered system and its corresponding standard system are given. In Section 5 recursive formulae for calculating the steadystate distributions of the orbit and system sizes have been developed. In Section 6, the relation between the continuous-time system and the discrete-time system has been analyzed. In Section 7, the busy period of an auxiliary system useful to study customers delay, has been obtained. In Section 8 we give the sojourn times distribution of a customer in the queue and in the system with its respective means. In Section 9 numerical results to illustrate the effect of the parameters on several performance characteristics are provided. Finally, a section of conclusions commenting the main research contributions of the paper is presented. In order to illustrate the approximation to the continuous-time model by its discrete-time counterpart, a brief study of the M/M/1 retrial queue subjected to the same discipline scheme that our system is included. 2. The mathematical model Let us consider a discrete-time retrial queue where the time axis is segmented into a sequence of equal time intervals (called slots). Further, let the time axis be marked by 0, 1, 2, . . . , m, . . .. It is assumed that all queueing activities (arrivals, departures and retrials) take place at the slot boundaries. For mathematical clarity it is supposed that a potential departure occurs in the interval (m− , m ), and a potential external arrival or a retrial, in this order, occur in the interval (m, m+ ); that is, external arrivals occur before retrials, and both take place at the moment immediately after the slot boundaries, and the departures at the moment immediately before the slot boundaries. Customers arrive according to a geometrical arrival process with probability a, i.e., a is the probability that an arrival occurs in a slot. If an arriving customer finds the server free, it immediately begins its service. Otherwise, with probability θ it expels out of the system the customer that is currently being served, or with complementary probability θ¯ it enters a group of blocked customers called ”orbit” in accordance with a FCFS discipline. It is assumed that only the customer at the head of the orbit is allowed for access to the server. The retrial time (the time between two successive attempts) follows a geometrical law with probability 1 − r, where r is the probability that the customer situated at the first place of the orbit does not make a retrial in a slot. The service times are independent and distributed with arbi i trary distribution {si }∞ and generating function S(x ) = ∞ i=1 si x . i=1 Hence, si is the probability that a service lasts i slots. It will be de noted by Sk = ∞ i=k si the probability that the service time lasts not less than k slots. 3. The Markov chain At time m+ (the instant immediately after time slot m), the state of the system can be described by the process
Xm = (Cm ,
ξm , Nm )
179
where Cm denotes the state of the server 0 or 1 according to whether the server is free or busy respectively and Nm is the number of repeated customers. If Cm = 1, then ξ m represents the remaining service time of the customer currently being served. It can be shown that {Xm , m ∈ N} is the Markov chain of our queueing system, whose space states is
χ = {(0, n ), (1, i, n ); i ≥ 1, n ≥ 0}. Our first task is to find the stationary distribution
π0,n = mlim P [Cm = 0, Nm = n], n ≥ 0 →∞ π1,i,n = mlim P [Cm = 1, ξm = i, Nm = n], i ≥ 1, n ≥ 0 →∞ of the Markov chain {Xm , m ∈ N}. The Kolmogorov equations for the stationary distribution of the system are given by
π0,0 = a¯ π0,0 + a¯ π1,1,0 ⇔ aπ0,0 = a¯ π1,1,0
(1)
π0,n = a¯ rπ0,n + a¯ rπ1,1,n , n ≥ 1
(2)
π1,i,n = asi π0,n + asi π1,1,n + a¯ (1 − r )si π1,1,n+1 ¯ 1,i+1,n−1 + a¯ π1,i+1,n +a¯ (1 − r )si π0,n+1 + (1 − δ0,n )aθπ +aθ si
∞
π1, j,n , i ≥ 1, n ≥ 0
(3)
j=2
where a¯ = 1 − a and δ a, b is the Kronecker’s delta. The normalization condition is ∞
π0,n +
∞ ∞
n=0
π1,i,n = 1
i=1 n=0
In order to solve (1)–(3) we introduce the following generating functions
ϕ0 ( z ) =
∞
π0,n zn
n=1
ϕ1 (x, z ) =
∞ ∞
π1,i,n xi zn
i=1 n=0
and the auxiliary generating functions
ϕ1,i (z ) =
∞
π1,i,n zn , i ≥ 1
n=0
Multiplying Eq. (2) by zn , summing over n and taking into account (1), we have
(1 − a¯ r )ϕ0 (z ) = a¯ rϕ1,1 (z ) − arπ0,0
(4)
Multiplying Eq. (3) by zn and summing over n, we get
a¯ (1 − r ) + aθ¯ z si ϕ1,1 ( z ) z a¯ (1 − r ) + az +aθ si ϕ1 (1, z ) + si ϕ0 ( z ) z 1−r−z − asi π0,0 , i ≥ 1 z
ϕ1,i (z ) = (a¯ + aθ¯ z )ϕ1,i+1 (z ) +
(5)
Then, multiplying the above equation by xi and summing over i, yields
z
x − (a¯ + aθ¯ z ) ϕ1 (x, z ) x = [[a¯ (1 − r ) + aθ¯ z]S(x ) − z(a¯ + aθ¯ z )]ϕ1,1 (z ) +[a¯ (1 − r ) + az]S(x )ϕ0 (z ) + aθ zS(x )ϕ1 (1, z ) −[1 − r − z]aS(x )π0,0
(6)
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I. Atencia / European Journal of Operational Research 256 (2017) 178–186
By substituting (4) into (6) we obtain
Corollary 2.
x − (a¯ + aθ¯ z ) ϕ1 (x, z ) x = [(a¯ [1 − r + arz] + aθ¯ z(1 − a¯ r ))S(x ) −(1 − a¯ r )z(a¯ + aθ¯ z )]ϕ1,1 (z )
1. The mean number of customers in the orbit is given by
(1 − a¯ r )z
E[N] = (1 ) = where
+(1 − a¯ r )aθ zS(x )ϕ1 (1, z ) −aS(x )(1 − r )(1 − z )π0,0
(7)
Choosing x = 1 in the above equation we have (8)
and inserting (8) into (7) leads to
[1 − a¯ r]θ¯ z
(1 − θ¯ z )(1 − r )aS(a¯ + aθ¯ z ) π0 , 0 D (z )
(9)
(10)
S(x ) − S(a¯ + aθ¯ z ) ax(a¯ + aθ¯ z )(1 − r )(1 − θ¯ z ) π0,0 (11) D (z ) x − (a¯ + aθ¯ z )
The normalization condition, that can be written as, π0,0 + ϕ0 (1 ) + ϕ1 (1, 1 ) = 1, allows us to find the unknown constant π 0, 0 :
(a¯ (1 − r ) + aθ¯ )S(a¯ + aθ¯ ) − (1 − a¯ r )θ¯ (a¯ + aθ¯ ) (1 − r )(a¯ + aθ¯ )θ Since π 0, 0 > 0, the inequality
π0 , 0 =
(12)
(a¯ (1 − r ) + aθ¯ )S(a¯ + aθ¯ ) − (1 − a¯ r )θ¯ (a¯ + aθ¯ ) > 0
(13)
is a necessary condition for the stability of the system. The previous results can be summarized in the following theorem Theorem 1. If condition (13) is fulfilled, the stationary distribution of the Markov chain {Xm , m ∈ N} has the following generating functions
ϕ1 (x, z ) =
D (z )
x − (a¯ + aθ¯ z )
π0 , 0
D(z ) = (a¯ (1 − r ) + aθ¯ z )S(a¯ + aθ¯ z ) − (1 − a¯ r )θ¯ z(a¯ + aθ¯ z )
where and
π0 , 0 =
aθ¯ rz[a¯ + aθ¯ z − S(a¯ + aθ¯ z )] π0 , 0 D (z ) S(x ) − S(a¯ + aθ¯ z ) ax(a¯ + aθ¯ z )(1 − r )(1 − θ¯ z )
(a¯ (1 − r ) + aθ¯ )S(a¯ + aθ¯ ) − (1 − a¯ r )θ¯ (a¯ + aθ¯ ) (1 − r )(a¯ + aθ¯ )θ
Corollary 1. 1. The probability generating function of the number of customers in the orbit (i.e., of the variable N) is given by
(z )=π0,0 +ϕ0 (z )+ϕ1 (1, z )=
(1−θ¯ z )(a¯ +aθ¯ z ) (1−r )π0,0 D (z )
2. The probability generating function of the number of customers in the system (i.e., of the variable L) is given by
(z ) = π0,0 + ϕ0 (z ) + z ϕ1 (1, z ) =
(a¯ + aθ¯ z )[(1 − z )S(a¯ + aθ¯ z ) + θ z] ( 1 − r ) π0 , 0 D (z )
θ
4. Stochastic decomposition
where D(z ) = (a¯ (1 − r ) + aθ¯ z )S(a¯ + aθ¯ z ) − (1 − a¯ r )θ¯ z(a¯ + aθ¯ z ) and substituting (10) into (9), we, finally obtain:
ϕ0 ( z ) =
1 − S(a¯ + aθ¯ )
ϕ1 (x, z )
Setting x = a¯ + aθ¯ z into Eq. (9) yields
ϕ1 (x, z ) =
2. The mean number of customers in the system is given by
E[L] = (1 ) = E[N] +
x = [(a¯ (1 − r ) + aθ¯ z )S(x ) − [1 − a¯ r]θ¯ z(a¯ + aθ¯ z )]ϕ1,1 (z ) −aS(x )(1 − r )(1 − θ¯ z )π0,0
ϕ1,1 ( z ) =
D(1 ) = (a¯ (1 − r ) + aθ¯ )S(a¯ + aθ¯ ) − (1 − a¯ r )θ¯ (a¯ + aθ¯ ) D (1 ) = θ¯ [a[S(a¯ + aθ¯ ) + [a¯ (1 − r ) + aθ¯ ]S (a¯ + aθ¯ )] −(1 − a¯ r )[a¯ + 2aθ¯ ]]
(1 − a¯ r )aθ¯ zϕ1 (1, z ) = [a¯ (1 − r ) + (1 − a¯ r )aθ¯ z]ϕ1,1 (z ) − a(1 − r )π0,0 x − (a¯ + aθ¯ z )
θ¯ [aθ − (a¯ + aθ¯ )]D(1 ) − θ (a¯ + aθ¯ )D (1 ) (a¯ + aθ¯ )θ D(1 )
This section discusses the analysis of the stochastic decomposition property for the system size distribution. Stochastic decomposition has been widely researched in M/G/1 vacation models. Fuhrmann and Cooper (1985) studied a general stochastic decomposition law for these queueing systems, which establishes that the number of customers in any vacation system in steady-state regime is distributed as the sum of two independent random variables: one being the number of customers in the standard M/G/1 system and the other being the number of customers in the vacation system given that the server is on vacation. This result was generalized by Shanthikumar (1988). The system under study satisfies this property, since it can be considered as a standard queue with server vacation. In this vacation model, the server begins a vacation when a service ends and there is neither an arrival nor a retrial. The duration of the vacation depends on the arrival process and the inter-retrial times. The vacation period finishes whenever an external customer arrives or the server chooses the customer at the head of the orbit. Under these considerations, the stochastic decomposition in this model is expressed in the following way
(z ) =
θ z + (1 − z )S(a¯ + aθ¯ z ) S(a¯ + aθ¯ ) − θ¯ π0,0 + ϕ0 (z ) · θ π0 , 0 + ϕ 0 ( 1 ) S(a¯ + aθ¯ z ) − θ¯ z
where the first fraction is the probability generating function of the number of customers in the Geo/G/1/∞ version of our model and the second fraction is the probability generating function of the number of customers in the orbit given that the server is idle. In fact, this is the stochastic decomposition law of our queueing system, i.e., the total number of customers in the system can be represented by the sum of two independent random variables: one is the total number of customers in the corresponding standard system and the other is the number of customers in the orbit given that the server is idle. This result can be summarized in the following theorem Theorem 2. The total number of customers in the system under study (L) can be represented as the sum of two independent random variables, one of which is the total number of customers in the corresponding Geo/G/1/∞ queueing system (L0 ) and the other is the number of repeated customers given that the server is idle (M). That is, L = L0 + M As an application of the stochastic decomposition property, we give a measure of the proximity between the steady-state distributions for the corresponding Geo/G/1/∞ queueing system and our queueing system. The importance of the following bounds is to provide upper and lower estimates for the distance between both distributions.
I. Atencia / European Journal of Operational Research 256 (2017) 178–186
Theorem 3. The following inequalities hold
S(a¯ + aθ¯ ) − θ¯ 2 1− θ
θ π0 , 0
S(a¯ + aθ¯ ) − θ¯
≤
∞
where dn =
| P[L = j] − P[L0 = j] |
j=0
≤ 2 1−
θ π0 , 0 S(a¯ + aθ¯ ) − θ¯
The proof of the preceding theorem follows the steps given in the paper by Artalejo and Falin (1994) and accordingly it is omitted. ∞ Finally, as we expected, the distance j=0 | P [L = j] − P [L0 = j] | between the distributions of the variables L and L0 decreases as r tends to 0. Therefore, for values of r close to 0, our system can be approximated by the corresponding model with infinite buffer.
In this section recursive formulae for some of the main stationary distributions of the system are given Theorem 4. The steady-state distribution of the number of customers in the orbit is given by the following recursive formulae
ψk = P [N = k] =
k−1
(14)
− a¯ k(r )ck−n ]ψn ,k≥1 k(r )S(a¯ )
n=0 [bk−n
(15)
Proof. The generating function (z) of the number of customers in the system satisfies the relation
1−z (z )G(z ) = 1 − [1 − S(a¯ + aθ¯ z )] k(r )π0,0 1 − θ¯ z
(19)
where G(z) and its expression in power series have been given in the proof of the previous theorem. Since ∞
1−
1−z [1 − S(a¯ + aθ¯ z )] = S(a¯ ) − (dn − dn−1 )zn 1 − θ¯ z n=1
φ0 g0 = k(r )S(a¯ )π0,0 ∞
φn gk−n = −(dk − dk−1 ), k ≥ 1
n=0
But g0 = k(r )S(a¯ ), gn = a¯ k(r )cn − bn , n ≥ 1, and consequently the formulae (17) and (18) are readily obtained. Let us note that if the service times are geometrically dis)x tributed with generating function S(x ) = ((11−s , then the coeffi−sx ) cients bn , cn and dn have the following closed-form expressions
(aθ¯ )n sn (1 − a¯ s )n (aθ¯ )n sn (1 − s ) cn = (1 − a¯ s )n+1 (aθ¯ )n+1 sn dn = (1 − a¯ s )n+1
1−r 1 − a¯ r ∞ i−1 bn = Si+1 a¯ i−n (aθ¯ )n , n ≥ 1 n−1
k (r ) =
i=n
∞ i=n
i si+1 a¯ i−n (aθ¯ )n , n ≥ 1 n
for n ≥ 1.
Proof. The generating function (z) of the number of customers in the orbit satisfies the relation
(z )G(z ) = k(r )π0,0 where
G(z ) = a¯ k(r )
(16)
S(a¯ + aθ¯ z ) S(a¯ + aθ¯ z ) θ¯ z − 1− 1 − θ¯ z a¯ + aθ¯ z a¯ + aθ¯ z
Using the properties of the generating functions and Newton’s binomial we get
G(z ) = k(r )S(a¯ ) −
∞ ∞ [bn − a¯ k(r )cn ] = gn z n n=0
i=n
Comparing the coefficients of zk on both sides of (16), yields
ψ 0 g 0 = k ( r ) π0 , 0 ,
∞
ψn gk−n = 0
n=0
Since g0 = k(r )S(a¯ ) and gn = a¯ k(r )cn − bn , Eqs. (14) and (15) are readily obtained. Theorem 5. The steady-state distribution of the number of customers in the system is given by the following recursive formulae
φ0 = P[L = 0] = π0,0 φk = P [ L = k ] = k≥1
i S a¯ i−n (aθ¯ )n a, n ≥ 0 n i+1
bn =
where
cn =
i=n
equating terms with equal coefficients power of zk on both sides of (19) leads to
5. Calculation of steady-state probabilities
π ψ0 = P[N = 0] = 0,0 S(a¯ )
∞
181
k−1
n=0 [bk−n
(17)
− a¯ k(r )cn ]φn − k(r )(dk − dk−1 )π0,0 , k(r )S(a¯ ) (18)
6. Relation to the continuous-time system This section deals with the analysis of the relation between the continuous-time system and the discrete-time system. We consider the M/G/1 retrial queue where the customers arrive according to a Poisson process with rate λ. If an arriving customer finds the server free, it immediately begins its service. Otherwise, with probability θ it expels out of the system the customer that is currently being served, or with complementary probability θ¯ it joins a group of blocked customers in accordance with a FCFS discipline, therefore only the customer at the head of the orbit may attempt retrials. The retrial time (the time between two consecutive attempts by the customer placed at the head of the orbit) is exponentially distributed with parameter μ. Customer service times are identically and independently distributed with distribution function B(x) and Laplace–Stieltjes transform β (s). If we suppose that time is divided into intervals of equal length , the continuous-time system can be approximated by a discretetime system for which
a=
λ
r = 1 − μ
i si = dB(x ), i ≥ 1 (i−1 )
where must be chosen sufficiently small so that a and r are probabilities.
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I. Atencia / European Journal of Operational Research 256 (2017) 178–186
We are going to obtain the probability generating function of the number of customers in the above described M/G/1 retrial queueing system, finding lim (z ). →0
First it is not difficult to prove that
lim S(a¯ + aθ¯ z ) = β (λ(1 − θ¯ z ))
→ 0
The proof of this equality follows using the definition of Lebesgue integration and will be omitted since the technique used can be found in Yang and Li (1995, Theorem 5). Taking into account the previous considerations, we get
lim (z )
→ 0
= lim (a¯ (1 − r )(1 − θ¯ z )S(a¯ + aθ¯ z ) − (1 − a¯ r )θ¯ z[(a¯ + aθ¯ z ) → 0
−S(a¯ + aθ¯ z )] )−1 (a¯ + aθ¯ z )[(1 − z )S(a¯ + aθ¯ z ) + θ z](1 − r )π0,0 = lim ((1 − λ)μ(1 − θ¯ z )S(a¯ + aθ¯ z ) − θ¯ z[(λ + μ ) → 0
−λμ2 ] · [(1 − λ(1 − θ¯ z )) − S(a¯ + aθ¯ z )] )−1 ·[1 − λ(1 − θ¯ z )][(1 − z )S(a¯ + aθ¯ z ) + θ z]μ π0,0 =
[(1 − z )β [λ(1 − θ¯ z )] + θ z]μ π0 , 0 ¯ μ(1 − θ z )β [λ(1 − θ¯ z )] − θ¯ z(λ + μ )[1 − β [λ(1 − θ¯ z )]]
where
π0 , 0 =
μθ β (λθ ) − (λ + μ )θ¯ [1 − β (λθ )]
In order to find the generating function of the sojourn time that a customer spends in the orbit, it is necessary to find the GF h(x; i) of the distribution of the busy period that starts with a customer in the server to which remains i slots to finish its service. This GF has the following expression
h ( x; i ) =
Let us explain the above formula: If after the first i − 1 slots no customer arrives to the system and in the slot i, either a new customer does not arrive and then the BP ends with probability 1 − aθ , or another customer arrives and then with probability aθ a new BP is opened with GF h(x). This accounts for the first term of the right hand side of the formula. Now, with respect to the second term, if after k − 1 slots, k = 1, . . . , i − 1, a new customer does not arrive (with probability (1 − aθ )k−1 ) and in the slot k a new customer arrives (with probability aθ ), a BP is opened with GF h(x). Summing over k from 1 to i − 1, the given formula of h(x; i) is obtained. Let us observe that h(x; i) can be written as
h ( x; i ) =
μθ
7. Busy period A busy period is defined as the period starting with the arrival of a customer that finds the system empty and ends if after the service completion epoch at which the system becomes empty and no external customer arrives. So, the busy period ends at the moment after a slot boundary. This section considers the busy period of an auxiliary system in which the arriving customers go directly to the server, that will be useful to study the customers delay in the original system. Specifically, we will suppose that the probability of an arrival is aθ , and as in the original model the arriving customer expels out of the system the customer that is currently being served, if any. Let’s denote by hk , k ≥ 0, the probability that the busy period of our auxiliary system lasts exactly k slots. Then we have:
h0 = 0 k
(1 − aθ )i−1 si aθ hk−i
i=1 k
(1 − aθ )i−1 Si+1 aθ hk−i , k ≥ 1
i=1
A recursive procedure of the above formula can lead to obtain numerically the distribution {hk , k ≥ 0} but in order to find the mo k ments of the distribution we will use the GF h(x ) = ∞ k=0 hk x , that is given by
aθ S[(1 − aθ )x]h(x ) 1 − aθ aθ (1 − aθ )x − S[(1 − aθ )x] + h (x ) 1 − aθ 1 − (1 − aθ )x
h(x ) = S[(1 − aθ )x] +
that is
h (x ) =
[(1 − aθ )x]i [1 − x(1 − aθ + aθ h(x ))] 1 − (1 − aθ )x aθ x + h (x ) 1 − (1 − aθ )x
8. Sojourn times 8.1. Sojourn time of a customer in the server In this section the distribution of the time that a customer spends in the server is obtained. With this aim let bk be the probability that the sojourn time of a customer in the server lasts exactly k slots. The distribution {bk , k ≥ 0} is given by
b0 = 0 bk = (1 − aθ )k−1 sk + aθ (1 − aθ )k−1 Sk+1 , k ≥ 1 The corresponding GF b(x ) =
∞
bk xk is given by
k=0
hk = (1 − aθ )k sk + +
[(1 − aθ )x]i [1 − aθ + aθ h(x )] 1 − aθ 1 − [(1 − aθ )x]i−1 +aθ x h ( x ), i ≥ 1 1 − (1 − aθ )x
[1 − (1 − aθ )x]S[(1 − aθ )x] 1 − x + aθ xS[(1 − aθ )x]
The mean length of a busy period is given by
1 − S(1 − aθ ) h¯ = h (1 ) = aθ S(1 − aθ )
b( x ) =
aθ x + (1 − x )S[(1 − aθ )x] 1 − (1 − aθ )x
and the mean time that a customer spends in the server is
1 − S(1 − aθ ) b¯ = b (1 ) = aθ Let us note that condition (13) can be written in the following way
(1 − a¯ r )θ¯ [(a¯ + aθ¯ ) − S(a¯ + aθ¯ )] < a¯ (1 − r )θ S(a¯ + aθ¯ ) which implies the inequality
aθ¯ [b¯ − 1] < a¯
1−r 1 − a¯ r
where the left hand side of the above inequality is the expected number of customers that enter in the orbit during a service time period and the second one the expected number of customers of the orbit that enter service at the epoch at which a service starts. Therefore, the condition (13) is also a sufficient condition for the stability of the system.
8.2. Sojourn time of a customer in the orbit The GF of the stationary distribution of the sojourn time of a customer in the orbit is given by
w (x ) =
π0 , 0 + ϕ 0 ( 1 ) + θ ϕ 1 ( 1 , 1 ) ¯ (x ) +θω
∞ ∞
π1,i,k h(x; i )(h(x )ω (x ))k
(20)
i=1 k=0
and using the GF’s obtained in Theorem 1, the above formula can be expressed in the form:
w (x ) =
π0 , 0 + ϕ 0 ( 1 ) + θ ϕ 1 ( 1 , 1 ) aθ h(x ) + θ¯ ϕ1 (1, h(x )ω (x )) 1 − (1 − aθ )x 1 − x(1 − aθ + aθ h(x )) + ϕ ((1 − aθ )x, h(x )ω (x )) ω (x ) 1 − (1 − aθ )x
Let us explain the formula (20): An arriving customer spends 0 slots in the orbit with probabil¯ ity π0,0 + ϕ0 (1 ) + θ ϕ1 (1, 1 ), and with probability θπ 1,i,k , i ≥ 1, k ≥ 0, it goes to the orbit finding k other customers before it and a customer in the server with a remaining service time of i slots. Then, the customer placed at the head of the orbit will wait there a period of time, counted since the arrival of the new customer till the beginning of its service, with GF h(x; i)ω(x), where ω(x) is the GF of the elapsed time from the ending of the BP h(x; i) until the beginning of its service. Once this customer begins its service a BP with GF h(x) is opened, and when this BP ends, the customer situated in the second place of the orbit at the moment of the arrival of the new customer, will wait a period of time till the beginning of his service with GF ω(x). Continuing in this way results the term
π1,i,k h(x; i )ω (x )(h(x )ω (x ))k and summing over i and k the formula (20) is obtained. Now, we are going to find the expression of the GF ω(x). Let us denote by ωk the probability that the customer placed at the head of the orbit spends exactly k slots in the orbit since the ending of a BP till the beginning of his service, then we have:
ω0 = 1 − r
+(1 − δ1,k )ar
k−1
(a¯ r )l−1
l=1
The GF ω (x ) =
183
0.7 0.6 0.5 0.4 0.3 0.2
θ = 0.1 θ = 0.3 θ = 0.7
0.1 0 0
0.2
0.4
r
0.6
0.8
1
Fig. 1. value for a = 0.2 and S (x ) = x2 .
no new customer arrives and a retrial occurs (all with probability a¯ (1 − r )). This account for the mentioned term. With respect to the second term let us make the following considerations: before the slot l, 1 ≤ l ≤ k − 1, no customer arrives and no retrial takes place (with probability a¯ l−1 r l ), in the slot l a new customer arrives to the system opening a BP with length of i slots, and ended this BP the customer placed at the head of the queue will wait there till the beginning of its service k − l − i slots (all with probability ahi ωk−l−i ). The mean sojourn time that a customer spends in the orbit is given by
w¯ = w (1 )=
θ¯
aθ 2
[(1+aθ h¯ )[1−S(1−aθ )−aθ (1−aθ )S (1−aθ )]
+aθ ω ¯ [1 − S(1 − aθ )] +
1 [aθ¯ [1 − S(1 − aθ ) D(1 )(1 − aθ )
−(1 − aθ )S (1 − aθ )]D(1 )
−[1 − S(1 − aθ )](1 − aθ )D (1 )](h¯ + ω ¯ )] 8.3. Sojourn time of a customer in the system The GF v(x ) of the stationary distribution of the sojourn time of a customer in the system is given by
ωk = (a¯ r )k (1 − r )
∞
The probability that the system is empty
I. Atencia / European Journal of Operational Research 256 (2017) 178–186
k−l
hi ωk−l−i , k ≥ 1
(21)
i=1
ωk xk is given by
v ( x ) = w ( x ) · b( x ) and its corresponding mean is given by
v¯ = v (1 ) = w¯ + b¯
k=0
ω (x ) =
(1 − r ) 1 − rx(a¯ + ah(x ))
and the corresponding mean time is given by
ω¯ = ω (1 ) =
r[1 + ah¯ ] 1−r
The explanation of the formula (21) is as follows: Consider the slot in which a BP ends, say the slot 0, then the customer at the first place of the orbit will begin immediately its service with probability 1 − r, (let us note that no customer arrives in the slot 0 since in this slot a BP has finished). Now, we consider the first term of the right hand side of the expression ωk , k ≥ 1: the customer at the head of the orbit will wait k, k ≥ 1, slots till the beginning of its service if in the slot 0 no retrial occurs (with probability r), in the following k − 1 slots no retrials occur and no customers arrive (all with probability (a¯ r )k−1 ), and in the slot k
9. Numerical results In this section we present some numerical results to illustrate the effect of varying parameters of the main performance measures of the system. In Fig. 1 the probability that the system is empty is plotted against the parameter r for three values of the parameter θ . As expected, π 0, 0 is decreasing as a function of r, and increases with increasing values of θ . Moreover, when the variable r approaches to the stability condition the probability that the system is empty tends to zero. Fig. 2 depicts the behavior of the mean numbers of repeated customers agains the parameter r for three values of the parameter θ . As expected, the mean orbit size is an increasing function of r and decreasing for increasing values of the parameter θ . When the parameter r approaches the stability condition, the mean orbit size tends to infinite due to the fact that the system becomes unstable.
I. Atencia / European Journal of Operational Research 256 (2017) 178–186
The mean number of customers in the orbit
184
Table 2 The steady-state distribution of the system size for a = 0.1, r = 0.6.
10 θ = 0.1 θ = 0.3 θ = 0.7
8
φ0 φ1 φ2 φ3 φ4 φ5 φ6 φ7 φ8 φ9 φ 10
6 4 2
θ = 0.3
θ = 0.5
θ = 0.7
0.7875 0.2153 0.0147 4.0921 · 10−4 1.1367 · 10−5 3.1575 · 10−7 8.7709 · 10−9 2.4363 · 10−10 6.7677 · 10−12 1.8799 · 10−13 5.2220 · 10−15
0.7925 0.2098 0.0113 2.4592 · 10−4 5.3132 · 10−6 1.1479 · 10−7 2.4800 · 10−9 5.3581 · 10−11 1.1576 · 10−12 2.5010 · 10−14 5.4035 · 10−16
0.7975 0.2042 0.0080 1.2462 · 10−4 1.9231 · 10−6 2.9678 · 10−8 4.5800 · 10−10 7.0680 · 10−12 1.0907 · 10−13 1.6832 · 10−15 2.5976 · 10−17
0.8025 0.1986 0.0048 4.4551 · 10−5 4.1250 · 10−7 3.8195 · 10−9 3.5366 · 10−11 3.2746 · 10−13 3.0320 · 10−15 2.8074 · 10−17 2.5995 · 10−19
0 0
0.2
0.4
r
0.6
0.8
1 0.9
Fig. 2. value for a = 0.2 and S (x ) = x2 .
Aproximation for differents Δ Continuous-time system
0.88
θ = 0.1
θ = 0.3
θ = 0.5
θ = 0.7
0.9722 0.0270 7.5017 · 10−4 2.0838 · 10−5 5.7883 · 10−7 1.6078 · 10−8 4.4663 · 10−10 1.2406 · 10−11 3.4462 · 10−13 9.5728 · 10−15 2.6591 · 10−16
0.9783 0.0211 4.5668 · 10−4 9.8667 · 10−6 2.1317 · 10−7 4.6055 · 10−9 9.9502 · 10−11 2.1497 · 10−12 4.6444 · 10−14 1.0034 · 10−15 2.1679 · 10−17
0.9845 0.0151 2.3447 · 10−4 3.6184 · 10−6 5.5840 · 10−8 8.6172 · 10−10 1.3298 · 10−11 2.0522 · 10−13 3.1669 · 10−15 4.8873 · 10−17 7.5421 · 10−19
0.9907 0.0091 8.4940 · 10−5 7.8648 · 10−7 7.2822 · 10−9 6.7428 · 10−11 6.2433 · 10−13 5.7808 · 10−15 5.3526 · 10−17 4.9561 · 10−19 4.5890 · 10−21
1 − π0,0
Table 1 The steady-state distribution of the orbit size for a = 0.1, r = 0.6.
ψ0 ψ1 ψ2 ψ3 ψ4 ψ5 ψ6 ψ7 ψ8 ψ9 ψ 10
θ = 0.1
0.86 0.84 0.82 0.8 −2 0.1 5 · 10
0.2
0.3
that the steady-state probabilities of the number of repeated customers in our discrete-time model tends to the values of the same probabilities in the corresponding M/M/1 when tends to zero. Fig. 4 depicts how the utilization factor of the discrete-time system tends, when approaches to zero, to the utilization factor of the M/M/1 system. Let us note that the approximation to any performance characteristic of the M/M/1 retrial queue can be carried out finding the corresponding characteristic in our discrete-time model with the service times geometrically distributed with parameter e−ν .
Utilization factor of works
Percentage (%)
80
74 65
82 75 65
85 76 66
88 67
100
85
81
78
100100
100 92
94
68
69
72
77
80 60 40 20
r=0.1 r=0.2 r=0.3 r=0.4 r=0.5 r=0.6 r=0.7 r=0.8 θ = 0.1
0.5
Fig. 4. Approximation of the Utilization factor of works for different values of and λ = 0.75, μ = 0.7, ν = 1, θ = 0.5.
Fig. 3, shows in percentages, how the utilization factor behaves for several values of r and θ . As it is expected, the utilization factor increases with increasing retrial rate r, and for fixed r decreases with increasing values of the parameter θ . Using the recursion scheme provide by Theorems 4 and 5 the formulae (14)–(18) have been implemented in Tables 1 and 2, summarizing our numerical experiments when the service always takes exactly two slots. Using the recursive formulae of Theorem 4, the approximation to the continuous-time retrial queue from the discrete-time retrial system has been showed. The M/M/1 retrial queue, described in the appendix, has been considered. In Table 3 it can be observed
79
0.4
θ = 0.3
θ = 0.7
Fig. 3. Utilization factor of work for a = 0.2 and different values of r and θ .
0
I. Atencia / European Journal of Operational Research 256 (2017) 178–186
185
Table 3 Approximation to the continuous-time of the steady-state distribution of the number of customers in the orbit for different values of and λ = 0.75, μ = 0.7, ν = 1, θ = 0.5
ψ0 ψ1 ψ2 ψ3 ψ4 ψ5 ψ6 ψ7 ψ8 ψ9 ψ 10
= 1/10
= 1/100
= 1/10 0 0
Continuous-time
.290997908834 .163375958509 .125729096601 .0967572332939 .0744613812378 .057303181447 .0440987603166 .0339370452447 .0261169028715 .0200987625964 .0154673875343
.284958781317 .160219809647 .124319239536 .096462936468 .0748484156332 .0580770763148 .0450637032827 .0349662462784 .0271313338615 .0210519960091 .0163348598424
.284179381272 .159843602528 .124150334409 .0964274158628 .0748950582709 .0581709019496 .0451812697893 .035092238066 .0272560992261 .0211697795851 .01644254241
.284090909090 .159801136363 .124131239853 .0964233738150 .074900299302 .0581814824941 .0451945444374 .0351064764826 .0272702094106 .0211831090957 .0164547365297
10. Conclusions and research results In this paper a discrete-time retrial queueing system in which the arriving customer may opt to follow a LCFS discipline expelling out of the system the customer that is in the server, if any, or to go to the last place of the orbit in accordance with a FCFS discipline has been studied. It is supposed that only the customer at the head of the orbit is allowed for access to the server. The retrial times follow a geometrical law with probability 1 − r, where r is the probability that the customer placed at the head of the orbit does not make a retrial in a slot. A thorough study of the system has been carried out obtaining the generating functions of the number of customers in the orbit and in the system, and its corresponding mean values. The stochastic decomposition property on the steady-state system size has been given, and as an application, we provide upper and lower estimates for the distance between the steady-state distributions for our queueing system and its corresponding standard system. An important feature of this paper is the recursive algorithm, provided by Theorems 4 and 5, for computing the steady-state probabilities of the number of customers in the orbit and in the system. The relation between the continuous-time system and the discrete-time system has also been analyzed, and the numerical examples show how the steady-state probabilities of the orbit size of the M/M/1 retrial queue, described in the appendix, can be approximated by the corresponding steady-state probabilities of the discrete-time model. The busy period of an auxiliary system has been considered to study the customer’s delay. One important research contribution of this paper is the analysis carried out to obtain the GF of the distribution of the sojourn time that a customer spends in the orbit, which provides, as far as the author knows, a new approach for the study of this subject in more general discrete-time retrial systems. In particular, a similar development to the one followed in Section 8.2 leads to obtain the sojourn time distribution of a customer in the orbit in discretetime queueing systems with general retrial times.
immediately its service, otherwise, with probability θ it expels out of the system the customer that is currently being served and begins his service, or with complementary probability θ¯ it joins a retrial group in accordance with a FCFS discipline. It is assumed that only the customer at the head of the orbit is allowed for access to the server. The time between two successive attempts by the customer at the head of the orbit is exponentially distributed with parameter μ. The service times are assumed to be independent and to have a common probability distribution ruled by an exponential law with parameter ν . We will denote by p0, n , n ≥ 0, the stationary probability that the server is free and there are n customers in the orbit, and by p1, n , n ≥ 0, the stationary probability that the server is busy and there are n customers in the orbit. Our objective is to find the stationary distribution of the orbit and system size. The set of statistical equilibrium equations is
λ p0,0 = ν p1,0
(1)
(λ + μ ) p0,n = ν p1,n , n ≥ 1
(2)
(ν + λθ¯ ) p1,n = λ p0,n + μ p0,n+1 + (1 − δ0,n )λθ¯ p1,n−1 , n ≥ 0
(3)
The normalization condition is
∞ n=0
p0,n +
∞
p1,n = 1.
n=0
In order to solve (1)–(3) we introduce the following generating functions
ϕ0 ( z ) =
∞
p0,n zn
n=1
ϕ1 ( z ) =
∞
p1,n zn
n=0
By multiplying (2) and (3) by zn and summing over n, these equations become
Acknowledgment
(λ + μ )ϕ0 (z ) = νϕ1 (z ) − λ p0,0
(4)
The author would like to thank the referees for valuable suggestions and comments that helped to improve the presentation of this paper. The research was supported by the National Spanish Project TIN2015-70266-C2-1-P.
z(ν + λθ¯ (1 − z ))ϕ1 (z ) = (μ + λz )ϕ0 (z ) − λzp0,0
(5)
Substituting (4) into (5) we get
ϕ1 ( z ) =
Appendix
λμ p0,0 νμ − λθ¯ (λ + μ )z
(6)
and from (6) and (4), we have A single-server continuous-time retrial queue in which external customers arrive according to a Poisson stream with rate λ is considered. If an arriving customer finds the server idle, it commences
ϕ0 ( z ) =
λ2 θ¯ z p0,0 νμ − λθ¯ (λ + μ )z
(7)
186
I. Atencia / European Journal of Operational Research 256 (2017) 178–186
The normalization condition, that can be written as π0,0 + ϕ0 (1 ) + ϕ1 (1 ) = 1, allows us to find the unknown constant p0, 0 :
νμ − λθ¯ (λ + μ ) p0,0 = (ν + λθ )μ
(8)
It will be assumed that the inequality νμ > λθ¯ (λ + μ ) is satisfied. The generating function of the number of customers in the orbit (i.e., of the variable N) is given by
ψ ∗ ( z ) = p0,0 + ϕ0 ( z ) + ϕ1 ( z ) =
λ + ν − λθ¯ z μ p0,0 νμ − λθ¯ (λ + μ )z
(9)
The generating function of the number of customers in the system (i.e., of the variable L) is given by
φ ∗ ( z ) = p0,0 + ϕ0 ( z ) + z ϕ1 ( z ) =
ν + λθ z μ p0,0 νμ − λθ¯ (λ + μ )z
(10)
From (9) and (10), the stationary distributions of the orbit and system size are readily obtained:
P [N = 0] = ψ0∗ =
λ+ν p ν 0,0
P [N = k] = ψk∗ = [(λ + ν )α − λθ¯ ]
α k−1 p , k≥1 ν 0,0
where
α=
λθ¯ (λ + μ ) νμ
and
P [L = 0] = ψ0∗ = p0,0 P [L = k] = ψk∗ = [να + λθ ]
α k−1 p , k≥1 ν 0,0
References Artalejo, J. (20 0 0). G-networks: A versatile approach for work removal in queueing networks. European Journal of Operational Research, 126, 233–249. Artalejo, J., & Falin, G. (1994). Stochastic decomposition for retrial queues. Top, 2, 329–342. Artalejo, J., & Gomez-Corral, A. (2008). Retrial queueing systems. Springer. Atencia, I. (2014). A discrete-time system with service control and repairs. International Journal of Applied Mathematics and Computer Science, 24(3), 471–484. Atencia, I. (2015). A discrete-time system queueing system with server breakdowns and changes in the repair times. Annals of Operations Research, 235, 37–49. Atencia, I., & Moreno, P. (2004). The discrete-time Geo/G/1 retrial queue with general retrial times. Queueing Systems, 48, 5–21. Choi, B., & Kim, J. (1997). Discrete-time Geo1 , Geo2 /G/1 retrial queueing systems with two types of calls,. Computers and Mathematics with Applications, 33, 79–88. Falin, G., & Templeton, J. (1997). Retrial queues. Chapman and Hall. Fiems, D., Steyaert, B., & Bruneel, H. (2002). Randomly interrupted GI/G/1 queues: service strategies and stability issues. Annals of Operations Research, 112, 171–183. Fuhrmann, S., & Cooper, R. (1985). Stochastic decomposition in the M/G/1 queue with generalized vacations. Operations Researc, 33(5), 1117–1129. Gelenbe, E., & Label, A. (1998). G-networks with multiple classes of signals and positive customers. European Journal of Operational Research, 108, 293–305. Krishnamoorthy, A., Gopakumar, B., & Viswanath Narayanan, V. (2012). A retrial queue with server interruptions, resumption and restart of service. Operations Research International Journal, 12, 133–149. Krishnamoorthy, A., Pramod, P., & Chakravarthy, S. (2013). A note on characterizing service interruptions with phase-type distribution. Journal of Stochastic Analysis and Applications, 31(4), 671–683. Krishnamoorthy, A., Pramod, P., & Chakravarthy, S. (2014). A survey on queues with interruptions. Top, 22, 290–320. Moreno, P., & Nobel, R. (2008). A discrete-time retrial queueing model with one server. European Journal of Operational Research, 189, 1088–1103. Shanthikumar, J. (1988). On stochastic decomposition in M/G/1 type queues with generalized server vacations. Operations Research, 36, 566–569. Shi, C., & Gershwin, S. (2016). Part sojourn time distribution in a two-machine line. European Journal of Operational Research, 248, 146–158. Walraevens, J., Steyaert, B., & Bruneel, H. (2006). A preemptive repeat priority queue with resampling: Performance analysis. Annals of Operations Research, 146, 189–202. Wu, J., Wang, J., & Liu, Z. (2013). A discrete-time Geo/G/1 retrial queue with preferred and impatient customers. Applied Mathematical Modelling, 37, 2552–2561. Yang, T., & Li, H. (1995). On the steady-state queue size distribution of the discrete– time Geo/G/1 queue with repeated customers. Queueing systems, 21, 199–215.
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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Stochastics and Statistics
A Geo/G/1 retrial queueing system with priority services I. Atencia University of Málaga, Higher Polytechnic School, Department of Applied Mathematics, 29071 Málaga, Spain
a r t i c l e
i n f o
Article history: Received 30 January 2016 Accepted 8 July 2016 Available online 14 July 2016 Keywords: Discrete-time Retrials Approximation to continuous-time Service upgrade Sojourn times
a b s t r a c t This paper considers a discrete-time retrial queueing system in which the arriving customers can decide to go directly to the server expelling out of the system the customer that is currently being served, if any, or to join the orbit in accordance with a FCFS discipline. An extensive analysis of the model has been carried out, and using a generating functions approach, the distributions of the number of customers in the orbit and in the system with its respective means are obtained. The stochastic decomposition law has been derived, and, as an application, bounds for the proximity between the steady-state distributions for the considered queueing system and its corresponding standard system are obtained. Also, recursive formulae for calculating the steady-state distributions of the orbit and system size have been developed. Besides, we prove that the M/G/1 retrial queue with service interruptions can be approximated by the corresponding discrete-time system. The generating function of the sojourn time of a customer in the orbit and in the system have also been provided. Finally, some numerical examples to illustrate the effect of the parameters on several performance characteristics and a section of conclusions commenting the main research contributions of this paper are presented. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Queueing systems that are characterized by the fact that a customer finding all the servers occupied upon arrival must abandon the service area, but some time later it returns to re-initiate its demand, are called retrial queueing systems. This type of customers repeat their attempts for service from a pool of unsatisfied customers (called orbit or retrial group) and are superimposed on the normal stream of external arrivals. Apart from theoretical interests, retrial queues have been successfully applied to telephone switching systems, telecommunication networks and computer systems: packet switch networks, shared bus local area networks operating under the Carrier-Sense Multiple Access protocol and collision avoidance star local area networks. A review of the main results, literature and discussion of practical situations where retrial queues arise can be located in the books by Falin and Templeton (1997) and Artalejo and Gomez-Corral (2008). At first, the study of the queues with repeated attempts was focused on the continuous-time, but Yang and Li (1995) extended the analysis to discrete-time systems giving insight to many researchers, for example, Choi and Kim (1997), Atencia and Moreno (2004), Moreno and Nobel (2008) and Wu, Wang, and Liu (2013). It is probably fair to say that the difficulty of the research on retrial models is partly due to their intractability, because from a practical point of
E-mail address: [email protected] http://dx.doi.org/10.1016/j.ejor.2016.07.011 0377-2217/© 2016 Elsevier B.V. All rights reserved.
view retrial models often describe a more realistic picture of many queueing situations than other types of models. Another feature that is usually found when a message is being processed in computers, in communications switching queues, etc., is that sometimes the information incoming to the server is more actual that the one on service. In that case, the message is moved to another place if the contained information can be used later on or if the information is not any more valuable it is deleted, in both cases the server is interrupted and upgraded. For a general survey on queues with service interruptions one can refer the papers Fiems, Steyaert, and Bruneel (2002), Walraevens, Steyaert, and Bruneel (2006), Krishnamoorthy, Gopakumar, and Viswanath Narayanan (2012), Krishnamoorthy, Pramod, and Chakravarthy (2013) and Atencia (2015), and for a detailed review on queues with service interruptions refer to the paper Krishnamoorthy, Pramod, and Chakravarthy (2014). The mechanism of moving messages by the arrival of one of them is called synchronized or triggered motion. There are several mechanisms on how and where the messages are moved, for a survey on them refer to Artalejo (20 0 0), Gelenbe and Label (1998) and Atencia (2014). In some manufacturing processes, the time that a part may spend in a buffer between successive operations is limited. Therefore, an important topic in delay models is the study of the sojourn-time distribution of an individual job. The probability distribution, rather than just the mean, or just the mean and
I. Atencia / European Journal of Operational Research 256 (2017) 178–186
standard deviation, is important because a factory’s profitability requires that no more than a specified percentage of the parts are scrapped. This issue is important in several industries, including the semiconductor, food, chemical, and steel industries, see for example, Shi and Gershwin (2016). The remainder of this paper is structured as follows. The assumptions of the queueing system under study are given in the next section. In Section 3, we study the Markov chain. The queue and system size distribution are obtained together with several performance measures of the system. In Section 4, a stochastic decomposition property, and, as an application, upper and lower estimates for the distance between the steady-state distribution for the considered system and its corresponding standard system are given. In Section 5 recursive formulae for calculating the steadystate distributions of the orbit and system sizes have been developed. In Section 6, the relation between the continuous-time system and the discrete-time system has been analyzed. In Section 7, the busy period of an auxiliary system useful to study customers delay, has been obtained. In Section 8 we give the sojourn times distribution of a customer in the queue and in the system with its respective means. In Section 9 numerical results to illustrate the effect of the parameters on several performance characteristics are provided. Finally, a section of conclusions commenting the main research contributions of the paper is presented. In order to illustrate the approximation to the continuous-time model by its discrete-time counterpart, a brief study of the M/M/1 retrial queue subjected to the same discipline scheme that our system is included. 2. The mathematical model Let us consider a discrete-time retrial queue where the time axis is segmented into a sequence of equal time intervals (called slots). Further, let the time axis be marked by 0, 1, 2, . . . , m, . . .. It is assumed that all queueing activities (arrivals, departures and retrials) take place at the slot boundaries. For mathematical clarity it is supposed that a potential departure occurs in the interval (m− , m ), and a potential external arrival or a retrial, in this order, occur in the interval (m, m+ ); that is, external arrivals occur before retrials, and both take place at the moment immediately after the slot boundaries, and the departures at the moment immediately before the slot boundaries. Customers arrive according to a geometrical arrival process with probability a, i.e., a is the probability that an arrival occurs in a slot. If an arriving customer finds the server free, it immediately begins its service. Otherwise, with probability θ it expels out of the system the customer that is currently being served, or with complementary probability θ¯ it enters a group of blocked customers called ”orbit” in accordance with a FCFS discipline. It is assumed that only the customer at the head of the orbit is allowed for access to the server. The retrial time (the time between two successive attempts) follows a geometrical law with probability 1 − r, where r is the probability that the customer situated at the first place of the orbit does not make a retrial in a slot. The service times are independent and distributed with arbi i trary distribution {si }∞ and generating function S(x ) = ∞ i=1 si x . i=1 Hence, si is the probability that a service lasts i slots. It will be de noted by Sk = ∞ i=k si the probability that the service time lasts not less than k slots. 3. The Markov chain At time m+ (the instant immediately after time slot m), the state of the system can be described by the process
Xm = (Cm ,
ξm , Nm )
179
where Cm denotes the state of the server 0 or 1 according to whether the server is free or busy respectively and Nm is the number of repeated customers. If Cm = 1, then ξ m represents the remaining service time of the customer currently being served. It can be shown that {Xm , m ∈ N} is the Markov chain of our queueing system, whose space states is
χ = {(0, n ), (1, i, n ); i ≥ 1, n ≥ 0}. Our first task is to find the stationary distribution
π0,n = mlim P [Cm = 0, Nm = n], n ≥ 0 →∞ π1,i,n = mlim P [Cm = 1, ξm = i, Nm = n], i ≥ 1, n ≥ 0 →∞ of the Markov chain {Xm , m ∈ N}. The Kolmogorov equations for the stationary distribution of the system are given by
π0,0 = a¯ π0,0 + a¯ π1,1,0 ⇔ aπ0,0 = a¯ π1,1,0
(1)
π0,n = a¯ rπ0,n + a¯ rπ1,1,n , n ≥ 1
(2)
π1,i,n = asi π0,n + asi π1,1,n + a¯ (1 − r )si π1,1,n+1 ¯ 1,i+1,n−1 + a¯ π1,i+1,n +a¯ (1 − r )si π0,n+1 + (1 − δ0,n )aθπ +aθ si
∞
π1, j,n , i ≥ 1, n ≥ 0
(3)
j=2
where a¯ = 1 − a and δ a, b is the Kronecker’s delta. The normalization condition is ∞
π0,n +
∞ ∞
n=0
π1,i,n = 1
i=1 n=0
In order to solve (1)–(3) we introduce the following generating functions
ϕ0 ( z ) =
∞
π0,n zn
n=1
ϕ1 (x, z ) =
∞ ∞
π1,i,n xi zn
i=1 n=0
and the auxiliary generating functions
ϕ1,i (z ) =
∞
π1,i,n zn , i ≥ 1
n=0
Multiplying Eq. (2) by zn , summing over n and taking into account (1), we have
(1 − a¯ r )ϕ0 (z ) = a¯ rϕ1,1 (z ) − arπ0,0
(4)
Multiplying Eq. (3) by zn and summing over n, we get
a¯ (1 − r ) + aθ¯ z si ϕ1,1 ( z ) z a¯ (1 − r ) + az +aθ si ϕ1 (1, z ) + si ϕ0 ( z ) z 1−r−z − asi π0,0 , i ≥ 1 z
ϕ1,i (z ) = (a¯ + aθ¯ z )ϕ1,i+1 (z ) +
(5)
Then, multiplying the above equation by xi and summing over i, yields
z
x − (a¯ + aθ¯ z ) ϕ1 (x, z ) x = [[a¯ (1 − r ) + aθ¯ z]S(x ) − z(a¯ + aθ¯ z )]ϕ1,1 (z ) +[a¯ (1 − r ) + az]S(x )ϕ0 (z ) + aθ zS(x )ϕ1 (1, z ) −[1 − r − z]aS(x )π0,0
(6)
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I. Atencia / European Journal of Operational Research 256 (2017) 178–186
By substituting (4) into (6) we obtain
Corollary 2.
x − (a¯ + aθ¯ z ) ϕ1 (x, z ) x = [(a¯ [1 − r + arz] + aθ¯ z(1 − a¯ r ))S(x ) −(1 − a¯ r )z(a¯ + aθ¯ z )]ϕ1,1 (z )
1. The mean number of customers in the orbit is given by
(1 − a¯ r )z
E[N] = (1 ) = where
+(1 − a¯ r )aθ zS(x )ϕ1 (1, z ) −aS(x )(1 − r )(1 − z )π0,0
(7)
Choosing x = 1 in the above equation we have (8)
and inserting (8) into (7) leads to
[1 − a¯ r]θ¯ z
(1 − θ¯ z )(1 − r )aS(a¯ + aθ¯ z ) π0 , 0 D (z )
(9)
(10)
S(x ) − S(a¯ + aθ¯ z ) ax(a¯ + aθ¯ z )(1 − r )(1 − θ¯ z ) π0,0 (11) D (z ) x − (a¯ + aθ¯ z )
The normalization condition, that can be written as, π0,0 + ϕ0 (1 ) + ϕ1 (1, 1 ) = 1, allows us to find the unknown constant π 0, 0 :
(a¯ (1 − r ) + aθ¯ )S(a¯ + aθ¯ ) − (1 − a¯ r )θ¯ (a¯ + aθ¯ ) (1 − r )(a¯ + aθ¯ )θ Since π 0, 0 > 0, the inequality
π0 , 0 =
(12)
(a¯ (1 − r ) + aθ¯ )S(a¯ + aθ¯ ) − (1 − a¯ r )θ¯ (a¯ + aθ¯ ) > 0
(13)
is a necessary condition for the stability of the system. The previous results can be summarized in the following theorem Theorem 1. If condition (13) is fulfilled, the stationary distribution of the Markov chain {Xm , m ∈ N} has the following generating functions
ϕ1 (x, z ) =
D (z )
x − (a¯ + aθ¯ z )
π0 , 0
D(z ) = (a¯ (1 − r ) + aθ¯ z )S(a¯ + aθ¯ z ) − (1 − a¯ r )θ¯ z(a¯ + aθ¯ z )
where and
π0 , 0 =
aθ¯ rz[a¯ + aθ¯ z − S(a¯ + aθ¯ z )] π0 , 0 D (z ) S(x ) − S(a¯ + aθ¯ z ) ax(a¯ + aθ¯ z )(1 − r )(1 − θ¯ z )
(a¯ (1 − r ) + aθ¯ )S(a¯ + aθ¯ ) − (1 − a¯ r )θ¯ (a¯ + aθ¯ ) (1 − r )(a¯ + aθ¯ )θ
Corollary 1. 1. The probability generating function of the number of customers in the orbit (i.e., of the variable N) is given by
(z )=π0,0 +ϕ0 (z )+ϕ1 (1, z )=
(1−θ¯ z )(a¯ +aθ¯ z ) (1−r )π0,0 D (z )
2. The probability generating function of the number of customers in the system (i.e., of the variable L) is given by
(z ) = π0,0 + ϕ0 (z ) + z ϕ1 (1, z ) =
(a¯ + aθ¯ z )[(1 − z )S(a¯ + aθ¯ z ) + θ z] ( 1 − r ) π0 , 0 D (z )
θ
4. Stochastic decomposition
where D(z ) = (a¯ (1 − r ) + aθ¯ z )S(a¯ + aθ¯ z ) − (1 − a¯ r )θ¯ z(a¯ + aθ¯ z ) and substituting (10) into (9), we, finally obtain:
ϕ0 ( z ) =
1 − S(a¯ + aθ¯ )
ϕ1 (x, z )
Setting x = a¯ + aθ¯ z into Eq. (9) yields
ϕ1 (x, z ) =
2. The mean number of customers in the system is given by
E[L] = (1 ) = E[N] +
x = [(a¯ (1 − r ) + aθ¯ z )S(x ) − [1 − a¯ r]θ¯ z(a¯ + aθ¯ z )]ϕ1,1 (z ) −aS(x )(1 − r )(1 − θ¯ z )π0,0
ϕ1,1 ( z ) =
D(1 ) = (a¯ (1 − r ) + aθ¯ )S(a¯ + aθ¯ ) − (1 − a¯ r )θ¯ (a¯ + aθ¯ ) D (1 ) = θ¯ [a[S(a¯ + aθ¯ ) + [a¯ (1 − r ) + aθ¯ ]S (a¯ + aθ¯ )] −(1 − a¯ r )[a¯ + 2aθ¯ ]]
(1 − a¯ r )aθ¯ zϕ1 (1, z ) = [a¯ (1 − r ) + (1 − a¯ r )aθ¯ z]ϕ1,1 (z ) − a(1 − r )π0,0 x − (a¯ + aθ¯ z )
θ¯ [aθ − (a¯ + aθ¯ )]D(1 ) − θ (a¯ + aθ¯ )D (1 ) (a¯ + aθ¯ )θ D(1 )
This section discusses the analysis of the stochastic decomposition property for the system size distribution. Stochastic decomposition has been widely researched in M/G/1 vacation models. Fuhrmann and Cooper (1985) studied a general stochastic decomposition law for these queueing systems, which establishes that the number of customers in any vacation system in steady-state regime is distributed as the sum of two independent random variables: one being the number of customers in the standard M/G/1 system and the other being the number of customers in the vacation system given that the server is on vacation. This result was generalized by Shanthikumar (1988). The system under study satisfies this property, since it can be considered as a standard queue with server vacation. In this vacation model, the server begins a vacation when a service ends and there is neither an arrival nor a retrial. The duration of the vacation depends on the arrival process and the inter-retrial times. The vacation period finishes whenever an external customer arrives or the server chooses the customer at the head of the orbit. Under these considerations, the stochastic decomposition in this model is expressed in the following way
(z ) =
θ z + (1 − z )S(a¯ + aθ¯ z ) S(a¯ + aθ¯ ) − θ¯ π0,0 + ϕ0 (z ) · θ π0 , 0 + ϕ 0 ( 1 ) S(a¯ + aθ¯ z ) − θ¯ z
where the first fraction is the probability generating function of the number of customers in the Geo/G/1/∞ version of our model and the second fraction is the probability generating function of the number of customers in the orbit given that the server is idle. In fact, this is the stochastic decomposition law of our queueing system, i.e., the total number of customers in the system can be represented by the sum of two independent random variables: one is the total number of customers in the corresponding standard system and the other is the number of customers in the orbit given that the server is idle. This result can be summarized in the following theorem Theorem 2. The total number of customers in the system under study (L) can be represented as the sum of two independent random variables, one of which is the total number of customers in the corresponding Geo/G/1/∞ queueing system (L0 ) and the other is the number of repeated customers given that the server is idle (M). That is, L = L0 + M As an application of the stochastic decomposition property, we give a measure of the proximity between the steady-state distributions for the corresponding Geo/G/1/∞ queueing system and our queueing system. The importance of the following bounds is to provide upper and lower estimates for the distance between both distributions.
I. Atencia / European Journal of Operational Research 256 (2017) 178–186
Theorem 3. The following inequalities hold
S(a¯ + aθ¯ ) − θ¯ 2 1− θ
θ π0 , 0
S(a¯ + aθ¯ ) − θ¯
≤
∞
where dn =
| P[L = j] − P[L0 = j] |
j=0
≤ 2 1−
θ π0 , 0 S(a¯ + aθ¯ ) − θ¯
The proof of the preceding theorem follows the steps given in the paper by Artalejo and Falin (1994) and accordingly it is omitted. ∞ Finally, as we expected, the distance j=0 | P [L = j] − P [L0 = j] | between the distributions of the variables L and L0 decreases as r tends to 0. Therefore, for values of r close to 0, our system can be approximated by the corresponding model with infinite buffer.
In this section recursive formulae for some of the main stationary distributions of the system are given Theorem 4. The steady-state distribution of the number of customers in the orbit is given by the following recursive formulae
ψk = P [N = k] =
k−1
(14)
− a¯ k(r )ck−n ]ψn ,k≥1 k(r )S(a¯ )
n=0 [bk−n
(15)
Proof. The generating function (z) of the number of customers in the system satisfies the relation
1−z (z )G(z ) = 1 − [1 − S(a¯ + aθ¯ z )] k(r )π0,0 1 − θ¯ z
(19)
where G(z) and its expression in power series have been given in the proof of the previous theorem. Since ∞
1−
1−z [1 − S(a¯ + aθ¯ z )] = S(a¯ ) − (dn − dn−1 )zn 1 − θ¯ z n=1
φ0 g0 = k(r )S(a¯ )π0,0 ∞
φn gk−n = −(dk − dk−1 ), k ≥ 1
n=0
But g0 = k(r )S(a¯ ), gn = a¯ k(r )cn − bn , n ≥ 1, and consequently the formulae (17) and (18) are readily obtained. Let us note that if the service times are geometrically dis)x tributed with generating function S(x ) = ((11−s , then the coeffi−sx ) cients bn , cn and dn have the following closed-form expressions
(aθ¯ )n sn (1 − a¯ s )n (aθ¯ )n sn (1 − s ) cn = (1 − a¯ s )n+1 (aθ¯ )n+1 sn dn = (1 − a¯ s )n+1
1−r 1 − a¯ r ∞ i−1 bn = Si+1 a¯ i−n (aθ¯ )n , n ≥ 1 n−1
k (r ) =
i=n
∞ i=n
i si+1 a¯ i−n (aθ¯ )n , n ≥ 1 n
for n ≥ 1.
Proof. The generating function (z) of the number of customers in the orbit satisfies the relation
(z )G(z ) = k(r )π0,0 where
G(z ) = a¯ k(r )
(16)
S(a¯ + aθ¯ z ) S(a¯ + aθ¯ z ) θ¯ z − 1− 1 − θ¯ z a¯ + aθ¯ z a¯ + aθ¯ z
Using the properties of the generating functions and Newton’s binomial we get
G(z ) = k(r )S(a¯ ) −
∞ ∞ [bn − a¯ k(r )cn ] = gn z n n=0
i=n
Comparing the coefficients of zk on both sides of (16), yields
ψ 0 g 0 = k ( r ) π0 , 0 ,
∞
ψn gk−n = 0
n=0
Since g0 = k(r )S(a¯ ) and gn = a¯ k(r )cn − bn , Eqs. (14) and (15) are readily obtained. Theorem 5. The steady-state distribution of the number of customers in the system is given by the following recursive formulae
φ0 = P[L = 0] = π0,0 φk = P [ L = k ] = k≥1
i S a¯ i−n (aθ¯ )n a, n ≥ 0 n i+1
bn =
where
cn =
i=n
equating terms with equal coefficients power of zk on both sides of (19) leads to
5. Calculation of steady-state probabilities
π ψ0 = P[N = 0] = 0,0 S(a¯ )
∞
181
k−1
n=0 [bk−n
(17)
− a¯ k(r )cn ]φn − k(r )(dk − dk−1 )π0,0 , k(r )S(a¯ ) (18)
6. Relation to the continuous-time system This section deals with the analysis of the relation between the continuous-time system and the discrete-time system. We consider the M/G/1 retrial queue where the customers arrive according to a Poisson process with rate λ. If an arriving customer finds the server free, it immediately begins its service. Otherwise, with probability θ it expels out of the system the customer that is currently being served, or with complementary probability θ¯ it joins a group of blocked customers in accordance with a FCFS discipline, therefore only the customer at the head of the orbit may attempt retrials. The retrial time (the time between two consecutive attempts by the customer placed at the head of the orbit) is exponentially distributed with parameter μ. Customer service times are identically and independently distributed with distribution function B(x) and Laplace–Stieltjes transform β (s). If we suppose that time is divided into intervals of equal length , the continuous-time system can be approximated by a discretetime system for which
a=
λ
r = 1 − μ
i si = dB(x ), i ≥ 1 (i−1 )
where must be chosen sufficiently small so that a and r are probabilities.
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I. Atencia / European Journal of Operational Research 256 (2017) 178–186
We are going to obtain the probability generating function of the number of customers in the above described M/G/1 retrial queueing system, finding lim (z ). →0
First it is not difficult to prove that
lim S(a¯ + aθ¯ z ) = β (λ(1 − θ¯ z ))
→ 0
The proof of this equality follows using the definition of Lebesgue integration and will be omitted since the technique used can be found in Yang and Li (1995, Theorem 5). Taking into account the previous considerations, we get
lim (z )
→ 0
= lim (a¯ (1 − r )(1 − θ¯ z )S(a¯ + aθ¯ z ) − (1 − a¯ r )θ¯ z[(a¯ + aθ¯ z ) → 0
−S(a¯ + aθ¯ z )] )−1 (a¯ + aθ¯ z )[(1 − z )S(a¯ + aθ¯ z ) + θ z](1 − r )π0,0 = lim ((1 − λ)μ(1 − θ¯ z )S(a¯ + aθ¯ z ) − θ¯ z[(λ + μ ) → 0
−λμ2 ] · [(1 − λ(1 − θ¯ z )) − S(a¯ + aθ¯ z )] )−1 ·[1 − λ(1 − θ¯ z )][(1 − z )S(a¯ + aθ¯ z ) + θ z]μ π0,0 =
[(1 − z )β [λ(1 − θ¯ z )] + θ z]μ π0 , 0 ¯ μ(1 − θ z )β [λ(1 − θ¯ z )] − θ¯ z(λ + μ )[1 − β [λ(1 − θ¯ z )]]
where
π0 , 0 =
μθ β (λθ ) − (λ + μ )θ¯ [1 − β (λθ )]
In order to find the generating function of the sojourn time that a customer spends in the orbit, it is necessary to find the GF h(x; i) of the distribution of the busy period that starts with a customer in the server to which remains i slots to finish its service. This GF has the following expression
h ( x; i ) =
Let us explain the above formula: If after the first i − 1 slots no customer arrives to the system and in the slot i, either a new customer does not arrive and then the BP ends with probability 1 − aθ , or another customer arrives and then with probability aθ a new BP is opened with GF h(x). This accounts for the first term of the right hand side of the formula. Now, with respect to the second term, if after k − 1 slots, k = 1, . . . , i − 1, a new customer does not arrive (with probability (1 − aθ )k−1 ) and in the slot k a new customer arrives (with probability aθ ), a BP is opened with GF h(x). Summing over k from 1 to i − 1, the given formula of h(x; i) is obtained. Let us observe that h(x; i) can be written as
h ( x; i ) =
μθ
7. Busy period A busy period is defined as the period starting with the arrival of a customer that finds the system empty and ends if after the service completion epoch at which the system becomes empty and no external customer arrives. So, the busy period ends at the moment after a slot boundary. This section considers the busy period of an auxiliary system in which the arriving customers go directly to the server, that will be useful to study the customers delay in the original system. Specifically, we will suppose that the probability of an arrival is aθ , and as in the original model the arriving customer expels out of the system the customer that is currently being served, if any. Let’s denote by hk , k ≥ 0, the probability that the busy period of our auxiliary system lasts exactly k slots. Then we have:
h0 = 0 k
(1 − aθ )i−1 si aθ hk−i
i=1 k
(1 − aθ )i−1 Si+1 aθ hk−i , k ≥ 1
i=1
A recursive procedure of the above formula can lead to obtain numerically the distribution {hk , k ≥ 0} but in order to find the mo k ments of the distribution we will use the GF h(x ) = ∞ k=0 hk x , that is given by
aθ S[(1 − aθ )x]h(x ) 1 − aθ aθ (1 − aθ )x − S[(1 − aθ )x] + h (x ) 1 − aθ 1 − (1 − aθ )x
h(x ) = S[(1 − aθ )x] +
that is
h (x ) =
[(1 − aθ )x]i [1 − x(1 − aθ + aθ h(x ))] 1 − (1 − aθ )x aθ x + h (x ) 1 − (1 − aθ )x
8. Sojourn times 8.1. Sojourn time of a customer in the server In this section the distribution of the time that a customer spends in the server is obtained. With this aim let bk be the probability that the sojourn time of a customer in the server lasts exactly k slots. The distribution {bk , k ≥ 0} is given by
b0 = 0 bk = (1 − aθ )k−1 sk + aθ (1 − aθ )k−1 Sk+1 , k ≥ 1 The corresponding GF b(x ) =
∞
bk xk is given by
k=0
hk = (1 − aθ )k sk + +
[(1 − aθ )x]i [1 − aθ + aθ h(x )] 1 − aθ 1 − [(1 − aθ )x]i−1 +aθ x h ( x ), i ≥ 1 1 − (1 − aθ )x
[1 − (1 − aθ )x]S[(1 − aθ )x] 1 − x + aθ xS[(1 − aθ )x]
The mean length of a busy period is given by
1 − S(1 − aθ ) h¯ = h (1 ) = aθ S(1 − aθ )
b( x ) =
aθ x + (1 − x )S[(1 − aθ )x] 1 − (1 − aθ )x
and the mean time that a customer spends in the server is
1 − S(1 − aθ ) b¯ = b (1 ) = aθ Let us note that condition (13) can be written in the following way
(1 − a¯ r )θ¯ [(a¯ + aθ¯ ) − S(a¯ + aθ¯ )] < a¯ (1 − r )θ S(a¯ + aθ¯ ) which implies the inequality
aθ¯ [b¯ − 1] < a¯
1−r 1 − a¯ r
where the left hand side of the above inequality is the expected number of customers that enter in the orbit during a service time period and the second one the expected number of customers of the orbit that enter service at the epoch at which a service starts. Therefore, the condition (13) is also a sufficient condition for the stability of the system.
8.2. Sojourn time of a customer in the orbit The GF of the stationary distribution of the sojourn time of a customer in the orbit is given by
w (x ) =
π0 , 0 + ϕ 0 ( 1 ) + θ ϕ 1 ( 1 , 1 ) ¯ (x ) +θω
∞ ∞
π1,i,k h(x; i )(h(x )ω (x ))k
(20)
i=1 k=0
and using the GF’s obtained in Theorem 1, the above formula can be expressed in the form:
w (x ) =
π0 , 0 + ϕ 0 ( 1 ) + θ ϕ 1 ( 1 , 1 ) aθ h(x ) + θ¯ ϕ1 (1, h(x )ω (x )) 1 − (1 − aθ )x 1 − x(1 − aθ + aθ h(x )) + ϕ ((1 − aθ )x, h(x )ω (x )) ω (x ) 1 − (1 − aθ )x
Let us explain the formula (20): An arriving customer spends 0 slots in the orbit with probabil¯ ity π0,0 + ϕ0 (1 ) + θ ϕ1 (1, 1 ), and with probability θπ 1,i,k , i ≥ 1, k ≥ 0, it goes to the orbit finding k other customers before it and a customer in the server with a remaining service time of i slots. Then, the customer placed at the head of the orbit will wait there a period of time, counted since the arrival of the new customer till the beginning of its service, with GF h(x; i)ω(x), where ω(x) is the GF of the elapsed time from the ending of the BP h(x; i) until the beginning of its service. Once this customer begins its service a BP with GF h(x) is opened, and when this BP ends, the customer situated in the second place of the orbit at the moment of the arrival of the new customer, will wait a period of time till the beginning of his service with GF ω(x). Continuing in this way results the term
π1,i,k h(x; i )ω (x )(h(x )ω (x ))k and summing over i and k the formula (20) is obtained. Now, we are going to find the expression of the GF ω(x). Let us denote by ωk the probability that the customer placed at the head of the orbit spends exactly k slots in the orbit since the ending of a BP till the beginning of his service, then we have:
ω0 = 1 − r
+(1 − δ1,k )ar
k−1
(a¯ r )l−1
l=1
The GF ω (x ) =
183
0.7 0.6 0.5 0.4 0.3 0.2
θ = 0.1 θ = 0.3 θ = 0.7
0.1 0 0
0.2
0.4
r
0.6
0.8
1
Fig. 1. value for a = 0.2 and S (x ) = x2 .
no new customer arrives and a retrial occurs (all with probability a¯ (1 − r )). This account for the mentioned term. With respect to the second term let us make the following considerations: before the slot l, 1 ≤ l ≤ k − 1, no customer arrives and no retrial takes place (with probability a¯ l−1 r l ), in the slot l a new customer arrives to the system opening a BP with length of i slots, and ended this BP the customer placed at the head of the queue will wait there till the beginning of its service k − l − i slots (all with probability ahi ωk−l−i ). The mean sojourn time that a customer spends in the orbit is given by
w¯ = w (1 )=
θ¯
aθ 2
[(1+aθ h¯ )[1−S(1−aθ )−aθ (1−aθ )S (1−aθ )]
+aθ ω ¯ [1 − S(1 − aθ )] +
1 [aθ¯ [1 − S(1 − aθ ) D(1 )(1 − aθ )
−(1 − aθ )S (1 − aθ )]D(1 )
−[1 − S(1 − aθ )](1 − aθ )D (1 )](h¯ + ω ¯ )] 8.3. Sojourn time of a customer in the system The GF v(x ) of the stationary distribution of the sojourn time of a customer in the system is given by
ωk = (a¯ r )k (1 − r )
∞
The probability that the system is empty
I. Atencia / European Journal of Operational Research 256 (2017) 178–186
k−l
hi ωk−l−i , k ≥ 1
(21)
i=1
ωk xk is given by
v ( x ) = w ( x ) · b( x ) and its corresponding mean is given by
v¯ = v (1 ) = w¯ + b¯
k=0
ω (x ) =
(1 − r ) 1 − rx(a¯ + ah(x ))
and the corresponding mean time is given by
ω¯ = ω (1 ) =
r[1 + ah¯ ] 1−r
The explanation of the formula (21) is as follows: Consider the slot in which a BP ends, say the slot 0, then the customer at the first place of the orbit will begin immediately its service with probability 1 − r, (let us note that no customer arrives in the slot 0 since in this slot a BP has finished). Now, we consider the first term of the right hand side of the expression ωk , k ≥ 1: the customer at the head of the orbit will wait k, k ≥ 1, slots till the beginning of its service if in the slot 0 no retrial occurs (with probability r), in the following k − 1 slots no retrials occur and no customers arrive (all with probability (a¯ r )k−1 ), and in the slot k
9. Numerical results In this section we present some numerical results to illustrate the effect of varying parameters of the main performance measures of the system. In Fig. 1 the probability that the system is empty is plotted against the parameter r for three values of the parameter θ . As expected, π 0, 0 is decreasing as a function of r, and increases with increasing values of θ . Moreover, when the variable r approaches to the stability condition the probability that the system is empty tends to zero. Fig. 2 depicts the behavior of the mean numbers of repeated customers agains the parameter r for three values of the parameter θ . As expected, the mean orbit size is an increasing function of r and decreasing for increasing values of the parameter θ . When the parameter r approaches the stability condition, the mean orbit size tends to infinite due to the fact that the system becomes unstable.
I. Atencia / European Journal of Operational Research 256 (2017) 178–186
The mean number of customers in the orbit
184
Table 2 The steady-state distribution of the system size for a = 0.1, r = 0.6.
10 θ = 0.1 θ = 0.3 θ = 0.7
8
φ0 φ1 φ2 φ3 φ4 φ5 φ6 φ7 φ8 φ9 φ 10
6 4 2
θ = 0.3
θ = 0.5
θ = 0.7
0.7875 0.2153 0.0147 4.0921 · 10−4 1.1367 · 10−5 3.1575 · 10−7 8.7709 · 10−9 2.4363 · 10−10 6.7677 · 10−12 1.8799 · 10−13 5.2220 · 10−15
0.7925 0.2098 0.0113 2.4592 · 10−4 5.3132 · 10−6 1.1479 · 10−7 2.4800 · 10−9 5.3581 · 10−11 1.1576 · 10−12 2.5010 · 10−14 5.4035 · 10−16
0.7975 0.2042 0.0080 1.2462 · 10−4 1.9231 · 10−6 2.9678 · 10−8 4.5800 · 10−10 7.0680 · 10−12 1.0907 · 10−13 1.6832 · 10−15 2.5976 · 10−17
0.8025 0.1986 0.0048 4.4551 · 10−5 4.1250 · 10−7 3.8195 · 10−9 3.5366 · 10−11 3.2746 · 10−13 3.0320 · 10−15 2.8074 · 10−17 2.5995 · 10−19
0 0
0.2
0.4
r
0.6
0.8
1 0.9
Fig. 2. value for a = 0.2 and S (x ) = x2 .
Aproximation for differents Δ Continuous-time system
0.88
θ = 0.1
θ = 0.3
θ = 0.5
θ = 0.7
0.9722 0.0270 7.5017 · 10−4 2.0838 · 10−5 5.7883 · 10−7 1.6078 · 10−8 4.4663 · 10−10 1.2406 · 10−11 3.4462 · 10−13 9.5728 · 10−15 2.6591 · 10−16
0.9783 0.0211 4.5668 · 10−4 9.8667 · 10−6 2.1317 · 10−7 4.6055 · 10−9 9.9502 · 10−11 2.1497 · 10−12 4.6444 · 10−14 1.0034 · 10−15 2.1679 · 10−17
0.9845 0.0151 2.3447 · 10−4 3.6184 · 10−6 5.5840 · 10−8 8.6172 · 10−10 1.3298 · 10−11 2.0522 · 10−13 3.1669 · 10−15 4.8873 · 10−17 7.5421 · 10−19
0.9907 0.0091 8.4940 · 10−5 7.8648 · 10−7 7.2822 · 10−9 6.7428 · 10−11 6.2433 · 10−13 5.7808 · 10−15 5.3526 · 10−17 4.9561 · 10−19 4.5890 · 10−21
1 − π0,0
Table 1 The steady-state distribution of the orbit size for a = 0.1, r = 0.6.
ψ0 ψ1 ψ2 ψ3 ψ4 ψ5 ψ6 ψ7 ψ8 ψ9 ψ 10
θ = 0.1
0.86 0.84 0.82 0.8 −2 0.1 5 · 10
0.2
0.3
that the steady-state probabilities of the number of repeated customers in our discrete-time model tends to the values of the same probabilities in the corresponding M/M/1 when tends to zero. Fig. 4 depicts how the utilization factor of the discrete-time system tends, when approaches to zero, to the utilization factor of the M/M/1 system. Let us note that the approximation to any performance characteristic of the M/M/1 retrial queue can be carried out finding the corresponding characteristic in our discrete-time model with the service times geometrically distributed with parameter e−ν .
Utilization factor of works
Percentage (%)
80
74 65
82 75 65
85 76 66
88 67
100
85
81
78
100100
100 92
94
68
69
72
77
80 60 40 20
r=0.1 r=0.2 r=0.3 r=0.4 r=0.5 r=0.6 r=0.7 r=0.8 θ = 0.1
0.5
Fig. 4. Approximation of the Utilization factor of works for different values of and λ = 0.75, μ = 0.7, ν = 1, θ = 0.5.
Fig. 3, shows in percentages, how the utilization factor behaves for several values of r and θ . As it is expected, the utilization factor increases with increasing retrial rate r, and for fixed r decreases with increasing values of the parameter θ . Using the recursion scheme provide by Theorems 4 and 5 the formulae (14)–(18) have been implemented in Tables 1 and 2, summarizing our numerical experiments when the service always takes exactly two slots. Using the recursive formulae of Theorem 4, the approximation to the continuous-time retrial queue from the discrete-time retrial system has been showed. The M/M/1 retrial queue, described in the appendix, has been considered. In Table 3 it can be observed
79
0.4
θ = 0.3
θ = 0.7
Fig. 3. Utilization factor of work for a = 0.2 and different values of r and θ .
0
I. Atencia / European Journal of Operational Research 256 (2017) 178–186
185
Table 3 Approximation to the continuous-time of the steady-state distribution of the number of customers in the orbit for different values of and λ = 0.75, μ = 0.7, ν = 1, θ = 0.5
ψ0 ψ1 ψ2 ψ3 ψ4 ψ5 ψ6 ψ7 ψ8 ψ9 ψ 10
= 1/10
= 1/100
= 1/10 0 0
Continuous-time
.290997908834 .163375958509 .125729096601 .0967572332939 .0744613812378 .057303181447 .0440987603166 .0339370452447 .0261169028715 .0200987625964 .0154673875343
.284958781317 .160219809647 .124319239536 .096462936468 .0748484156332 .0580770763148 .0450637032827 .0349662462784 .0271313338615 .0210519960091 .0163348598424
.284179381272 .159843602528 .124150334409 .0964274158628 .0748950582709 .0581709019496 .0451812697893 .035092238066 .0272560992261 .0211697795851 .01644254241
.284090909090 .159801136363 .124131239853 .0964233738150 .074900299302 .0581814824941 .0451945444374 .0351064764826 .0272702094106 .0211831090957 .0164547365297
10. Conclusions and research results In this paper a discrete-time retrial queueing system in which the arriving customer may opt to follow a LCFS discipline expelling out of the system the customer that is in the server, if any, or to go to the last place of the orbit in accordance with a FCFS discipline has been studied. It is supposed that only the customer at the head of the orbit is allowed for access to the server. The retrial times follow a geometrical law with probability 1 − r, where r is the probability that the customer placed at the head of the orbit does not make a retrial in a slot. A thorough study of the system has been carried out obtaining the generating functions of the number of customers in the orbit and in the system, and its corresponding mean values. The stochastic decomposition property on the steady-state system size has been given, and as an application, we provide upper and lower estimates for the distance between the steady-state distributions for our queueing system and its corresponding standard system. An important feature of this paper is the recursive algorithm, provided by Theorems 4 and 5, for computing the steady-state probabilities of the number of customers in the orbit and in the system. The relation between the continuous-time system and the discrete-time system has also been analyzed, and the numerical examples show how the steady-state probabilities of the orbit size of the M/M/1 retrial queue, described in the appendix, can be approximated by the corresponding steady-state probabilities of the discrete-time model. The busy period of an auxiliary system has been considered to study the customer’s delay. One important research contribution of this paper is the analysis carried out to obtain the GF of the distribution of the sojourn time that a customer spends in the orbit, which provides, as far as the author knows, a new approach for the study of this subject in more general discrete-time retrial systems. In particular, a similar development to the one followed in Section 8.2 leads to obtain the sojourn time distribution of a customer in the orbit in discretetime queueing systems with general retrial times.
immediately its service, otherwise, with probability θ it expels out of the system the customer that is currently being served and begins his service, or with complementary probability θ¯ it joins a retrial group in accordance with a FCFS discipline. It is assumed that only the customer at the head of the orbit is allowed for access to the server. The time between two successive attempts by the customer at the head of the orbit is exponentially distributed with parameter μ. The service times are assumed to be independent and to have a common probability distribution ruled by an exponential law with parameter ν . We will denote by p0, n , n ≥ 0, the stationary probability that the server is free and there are n customers in the orbit, and by p1, n , n ≥ 0, the stationary probability that the server is busy and there are n customers in the orbit. Our objective is to find the stationary distribution of the orbit and system size. The set of statistical equilibrium equations is
λ p0,0 = ν p1,0
(1)
(λ + μ ) p0,n = ν p1,n , n ≥ 1
(2)
(ν + λθ¯ ) p1,n = λ p0,n + μ p0,n+1 + (1 − δ0,n )λθ¯ p1,n−1 , n ≥ 0
(3)
The normalization condition is
∞ n=0
p0,n +
∞
p1,n = 1.
n=0
In order to solve (1)–(3) we introduce the following generating functions
ϕ0 ( z ) =
∞
p0,n zn
n=1
ϕ1 ( z ) =
∞
p1,n zn
n=0
By multiplying (2) and (3) by zn and summing over n, these equations become
Acknowledgment
(λ + μ )ϕ0 (z ) = νϕ1 (z ) − λ p0,0
(4)
The author would like to thank the referees for valuable suggestions and comments that helped to improve the presentation of this paper. The research was supported by the National Spanish Project TIN2015-70266-C2-1-P.
z(ν + λθ¯ (1 − z ))ϕ1 (z ) = (μ + λz )ϕ0 (z ) − λzp0,0
(5)
Substituting (4) into (5) we get
ϕ1 ( z ) =
Appendix
λμ p0,0 νμ − λθ¯ (λ + μ )z
(6)
and from (6) and (4), we have A single-server continuous-time retrial queue in which external customers arrive according to a Poisson stream with rate λ is considered. If an arriving customer finds the server idle, it commences
ϕ0 ( z ) =
λ2 θ¯ z p0,0 νμ − λθ¯ (λ + μ )z
(7)
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I. Atencia / European Journal of Operational Research 256 (2017) 178–186
The normalization condition, that can be written as π0,0 + ϕ0 (1 ) + ϕ1 (1 ) = 1, allows us to find the unknown constant p0, 0 :
νμ − λθ¯ (λ + μ ) p0,0 = (ν + λθ )μ
(8)
It will be assumed that the inequality νμ > λθ¯ (λ + μ ) is satisfied. The generating function of the number of customers in the orbit (i.e., of the variable N) is given by
ψ ∗ ( z ) = p0,0 + ϕ0 ( z ) + ϕ1 ( z ) =
λ + ν − λθ¯ z μ p0,0 νμ − λθ¯ (λ + μ )z
(9)
The generating function of the number of customers in the system (i.e., of the variable L) is given by
φ ∗ ( z ) = p0,0 + ϕ0 ( z ) + z ϕ1 ( z ) =
ν + λθ z μ p0,0 νμ − λθ¯ (λ + μ )z
(10)
From (9) and (10), the stationary distributions of the orbit and system size are readily obtained:
P [N = 0] = ψ0∗ =
λ+ν p ν 0,0
P [N = k] = ψk∗ = [(λ + ν )α − λθ¯ ]
α k−1 p , k≥1 ν 0,0
where
α=
λθ¯ (λ + μ ) νμ
and
P [L = 0] = ψ0∗ = p0,0 P [L = k] = ψk∗ = [να + λθ ]
α k−1 p , k≥1 ν 0,0
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