AA Polling Model with Reneging at Polling Instants ∗ M.A.A. Boon † [email protected], 2010 Abstract
In this paper we consider a single-server, cyclic polling system with switch-over timesand Poisson arrivals. The service disciplines that are discussed, are exhaustive and gatedservice. The novel contribution of the present paper is that we consider the reneging ofcustomers at polling instants. In more detail, whenever the server starts or ends a visitto a queue, some of the customers waiting in each queue leave the system before havingreceived service. The probability that a certain customer leaves the queue, depends onthe queue in which the customer is waiting, and on the location of the server. We showthat this system can be analysed by introducing customer subtypes, depending on theirarrival periods, and keeping track of the moment when they abandon the system. In orderto determine waiting time distributions, we regard the system as a polling model withvarying arrival rates, and apply a generalised version of the distributional form of Little’slaw. The marginal queue length distribution can be found by conditioning on the stateof the system (position of the server, and whether it is serving or switching).
Keywords:
Polling, reneging, varying arrival rates, queue lengths, waiting times
A polling system is a queueing system that consists of multiple queues being served by oneserver, generally in a fixed, cyclic, order. There is a vast literature on polling systems, moti-vated by many real-life applications. These applications are frequently found in productionenvironments, where one machine produces different part types. Typically, after the produc-tion of several parts of the same type, the machine is reconfigured and starts producing partsof the next type. The performance measures of interest are, e.g., the mean throughput of themachine, the number of different product orders that are waiting to be processed, and themean order lead time (i.e., the time between the placement of the order and the completionof the last item in the order). Other typical application areas of polling systems are telecom-munications, where several protocols use a round-robin principle for the communication ofdata packets between multiple devices, and transportation. We recommend surveys of, e.g., ∗ The research was done in the framework of the BSIK/BRICKS project, and of the European Network ofExcellence Euro-NF. † Eurandom and Department of Mathematics and Computer Science, Eindhoven University of Technology,P.O. Box 513, 5600MB Eindhoven, The Netherlands a r X i v : . [ m a t h . P R ] A ug akagi [24], Levy and Sidi [18] and Vishnevskii and Semenova [25], for a better overview ofapplications of polling systems, and techniques to analyse them.In queueing systems, waiting is an inevitable nuisance. When waiting times become too large,the impatience grows and customers might decide to leave the queue and possibly returnanother time. This phenomenon, which occurs in many real-life situations, is referred to inqueueing literature as reneging . Alternatively, the terms abandonment or impatience are used.The first paper on the subject of abandonment due to impatience, has been written by Palm[21] who studies annoyance of customers in telecommunications. In [21], as in most of theliterature on reneging, impatience is modelled as a timer that starts running at the momentthat a customer joins the queue. When this timer reaches a certain (usually random) valuewhile the customer is still waiting in the queue, he abandons the system immediately. Thesystem studied in [21] is an M/M/n queue with exponentially distributed customer patience,which is used frequently to model call centers, and is referred to as the Erlang-A (
M/M/n + M )queueing system. This model, and generalisations to other patience distributions, are studiedin more detail in by, e.g., Riordan [23], Haugen and Skogan [15], Baccelli and Hebuterne[4], and Boxma and De Waal [8]. The vast majority of papers on reneging focusses on theapplication to call centers, studying the loss probability and comparing different staffing rules.See [17, 20, 28] for some recent developments in reneging in queueing models for call centers.In the application area of computer systems, processor sharing is an important discipline andreneging has been studied in this context as well. Assaf and Haviv [3] consider a model wherecustomers may decide to abandon the system, depending on the number of customers that arein service simultaneously. Gromoll et al. [14] consider an overloaded processor sharing queuewith impatient customers, and find a scaling procedure that makes the model analyticallytractable.One common aspect of all models considered in the aforementioned literature, is that whenevercustomers are available in the system, (at least) one of them is in service. In polling models,customer impatience might be increased by absence of the server at the queue of arrival. Thiskind of behaviour has been studied in single-queue systems by Altman and Yechiali [2], whodiscuss M/G/
M/M/c queues with server vacations, and customers growing impatientwhile the server is away. Zhang et al. [29] study a similar system, but with an
M/M/ /N queue. Their model also includes balking, which means that customers may decide not to enterthe system at all, depending on the number of customers present in the queue. Madan [19]studies a system where server vacations may start at arbitrary moments, even when customersare being served or when the system is idle. Whenever a vacation starts, a random numberof customers abandons the system immediately. The fact that more than one customer canleave the system at the same time, and only at specified moments (in this case the beginningof a vacation) makes this model quite different from most of the other papers dealing withcustomer impatience. We refer to this kind of abandonment as synchronised reneging , a termthat is introduced by Adan et al. [1], who consider a model where each customer has thesame probability of abandoning the system at synchronised reneging epochs. They considera queueing system with server vacations that start as soon as the queue becomes empty,distinguishing between two cases. In the first case, which is called the Unique AbandonmentEpoch (UAE) model, customers leave the queue at visit beginnings only. In the second case,referred to as the Multiple Abandonment Epochs (MAE) model, impatient customers alsoabandon at (randomly) specified, synchronised moments during the server vacation.2n the present paper we study synchronised reneging in polling systems , with abandonmentstaking place at visit beginnings and endings. Basically, this means that we extend the UAEmodel of [1] to systems with multiple queues, thereby increasing the number of synchronisedreneging epochs. The higher frequency of abandonment epochs significantly increases thecomplexity of the analysis. Although the reneging policy considered in the current paper isbased on [1], the analysis is different. It is based on new techniques, developed in a recentpaper on polling models with so-called smart customers , cf. [5], to find waiting time and queuelength distributions in polling systems with varying arrival rates. In the present paper weuse and extend these techniques, so they can be applied to a polling model with synchronisedreneging. This makes it possible to extend the results of [1] in three new directions. Firstly,this allows us to study systems with multiple queues. Secondly, we can consider other servicedisciplines than exhaustive service (i.e., serve customers in a queue until it is empty). In thepresent paper we discuss not only exhaustive, but also gated service (i.e., serve all customerspresent at the server’s arrival at the queue). The third new contribution of the present paperis that we can compute other relevant performance measures as well, like the distributionsof the cycle times and the waiting times. These extensions provide the main motivation tostudy this model, which is one of the first attempts to introduce reneging in polling systems.The only related work is by Vishnevsky and Semenova [26] who study a two-queue pollingsystem with exponential service times and exhaustive service, with the more conventionaltimer to model the patience of the customer. They illustrate how the Power-series algorithmcan be used to find the equilibrium state probabilities, but no explicit performance measuresare computed.Since the analysis in the present paper relies heavily on [5], we briefly summarise their results.In a polling system with smart customers, the arrival rates of the different customer typesdepend on the state of the server, where state is defined as a combination of its location, andwhether it is working or switching. In this situation, it is no longer possible to apply thedistributional form of Little’s law in its standard form (see, for example, [16]), but it requiresa generalization, developed in [5]. In the present paper we apply this model with smartcustomers to determine the Laplace-Stieltjes Transform (LST) of the waiting time distributionof each customer type. This generalised version of the distributional form of Little’s law isapplied to the joint queue length distribution at departure epochs of customers that have notabandoned the system prematurely, which means that this waiting time is determined onlyfor customers that are actually served. To determine the Probability Generating Function(PGF) of the marginal queue length distribution of each customer type, we do need to takeinto account the impatient customers that abandon the system before being served. Thisrequires a different approach, as will be shown later in this paper.The structure of the present paper is as follows. In the next section we describe the modeland the notation in more detail. In Section 3 we introduce an alternative model with smartcustomers that is used to determine the cycle time and waiting time of served customers inthe original model. The stability condition is also presented in this section. The marginalqueue length distributions are studied in Section 4, because this requires a different approach.Section 5 discusses a special case of the model under consideration, a polling system withonly one queue and exhaustive service. This queueing system with server vacations, has beenstudied in [1]. We show that the queue length PGF obtained in Section 4 agrees with theirresult, and we mention some further results that have not been discussed in [1], like the cycletime and sojourn times of all customers, which reduce to elegant expressions when the system3onsists of only one queue. The last section discusses a numerical example to illustrate typicalfeatures of a polling model with synchronised reneging at polling instants. The polling model under consideration contains N queues, Q , . . . , Q N . These queues areserved by one server in a fixed, cyclic order. The time that is required to switch from Q i to Q i +1 is denoted by S i , which is called a switch-over time, with LST σ i ( · ). Throughoutthe paper, all indices are modulo N , so Q N +1 refers to Q and so on. The arrival process ofcustomers in Q i , denoted by type i customers, is a Poisson process with parameter λ i . Theservice time of a type i customer is denoted by B i , with LST β i ( · ). The switch-over times,interarrival times and service times are all assumed to be independent of each other. Theservice discipline of each queue determines when the server switches to the next queue. Thefollowing property, which is defined in [22] and [12], plays a key role in the analysis of pollingsystems. Property 2.1 (Branching Property)
If the server arrives at Q i to find k i customers there,then during the course of the server’s visit, each of these k i customers will effectively bereplaced in an i.i.d. manner by a random population having probability generating function h i ( z , . . . , z N ), which can be any N -dimensional probability generating function.Performance measures, like queue length distributions and waiting times, can be determinedfor polling systems with all queues satisfying Property 2.1, whereas only very few, exceptional,polling systems can be analysed if the service disciplines do not satisfy this property. In thepresent paper we discuss the two most common service disciplines satisfying this branchingproperty , exhaustive and gated service. A queue with exhaustive service is served until it iscompletely empty. In a queue with gated service only those customers are served, that arepresent at the beginning of a visit to this queue. The PGF h i ( z , . . . , z N ) in Property 2.1 is β i (cid:0) (cid:80) Nj =1 λ j (1 − z j ) (cid:1) for gated service, and π i (cid:0) (cid:80) j (cid:54) = i λ j (1 − z j ) (cid:1) for exhaustive service, where π i ( · ) is the LST of a busy period distribution in an M/G/ i customers,so it is the root in (0 ,
1] of the equation π i ( ω ) = β i ( ω + λ i (1 − π i ( ω ))), ω ≥ θ i ( · ) as the LST of the time that the server spends at Q i due to the presence ofone customer there. For gated service θ i ( · ) = β i ( · ), and for exhaustive service θ i ( · ) = π i ( · ).A cycle consists of the visit times of all queues, denoted by V , . . . , V N , and the switch-overtimes S , . . . , S N . The distribution of the length of one cycle depends on the starting point ofthis cycle. We use the notation C i for the time between two successive visit beginnings to Q i ,with LST γ i ( · ), and C ∗ i for the time between two successive visit completions to Q i , with LST γ ∗ i ( · ). When studying a queue with gated service, it turns out that C i plays an importantrole, whereas C ∗ i is used in the analysis of queues receiving exhaustive service. The intervisittime I i is the time between a visit completion of Q i and the next visit beginning at Q i .The model discussed in the present paper is different from models in existing literature becausecustomers grow impatient and may decide to leave the waiting line before actually beingserved. This is called reneging . The moments at which customers are allowed to leave thesystem, are only those moments when a new visit or switch-over time starts. For this reason,we refer to this model as a polling model with synchronised reneging at polling instants .4ow we give a more formal description. As stated before, a cycle consists of the periods V , S , . . . , V N , S N . Now let P ∈ { V , S , . . . , V N , S N } . At the moment that P starts, eachcustomer waiting in Q i immediately leaves the system with probability p ( P ) i , i = 1 , . . . , N . Wedenote the probability that a customer stays by q ( P ) i = 1 − p ( P ) i . The difficulty in the analysisof this model, is that customers in a certain queue may leave the system at more than just oneoccasion. We use different ways to circumvent this problem in order to find the performancemeasures of interest. An important part in the analysis, is the fact that we artificially spliteach visit time and switch-over time into two parts, a and b . Visit time V i is split into V ia and V ib . We consider V ia as the subperiod in which all the customers abandon the system,right before the start of the actual service of the type i customers, which takes place during V ib . So during V ia , first each type 1 customer abandons the system with probability p ( V i )1 ,followed by the type 2 customers, and so on, until the reneging of the type N customers. Thisrequires no time, so E [ V ia ] = 0. During V ib the service of the type i customers that remain inthe system takes place, so E [ V ib ] = E [ V i ]. Similarly, the switch-over time S i is also split into S ia and S ib , with E [ S ia ] = 0, and E [ S ib ] = E [ S i ]. During S ia the reneging of the customersthat abandon the system before the beginning of S i takes place. We need this way of lookingat the system, because the queue lengths at the beginning of V ia are different from the queuelengths at the beginning of V ib (and the same holds for the switch-over times). In fact, onecan regard V ia , for i = 1 , . . . , N , as separate visit periods during which subsequently type1 , . . . , N customers are served with probability p ( V i )1 , . . . , p ( V i ) N (or probability p ( S i )1 , . . . , p ( S i ) N for S ia ), with all service times equal to 0. In the present section we study the LSTs of the cycle time distribution, visit time distributions,and of the waiting time distribution of each customer type. The section ends with a note onthe stability condition of the model. The waiting time of a customer is the time between themoment of arrival, and the moment that the customer is taken into service. The waiting timeis only determined for customers that have not prematurely abandoned the system. The timethat all , including reneging, customers spend in the system, requires a different approach andis not discussed in detail in the present paper. We only show in an example in Section 5 howthis can be done.In this section we introduce a different way of looking at the system. Obviously, the length ofa visit V i is solely determined by those type i customers that have not abandoned the systemat any of the moments where this had been possible. A logical consequence is that onlycustomers that are eventually served, contribute to the cycle time and determine whether thesystem is stable or not. This observation forms the basis of the analysis in this section. If weremove reneging customers from the system and focus on the remaining customers only, wecan show that the system can be viewed as a polling system where the arrival rates of the N customer types depend on the state of the server, i.e., its location and whether it is servingor switching. This type of model is called a polling system with smart customers, introducedin [7], and analysed in more detail in [5]. The present section uses results from these papersand applies them to a polling model with reneging at polling instants.We start by introducing the joint queue length PGF at the beginning of all subperiods, denoted5y (cid:102) LB ( P ) ( z , . . . , z N ), where subperiod P ∈ { V a , V b , S a , S b , . . . , V Na , V Nb , S Na , S Nb } . ThePGFs of the joint queue length distributions at the beginnings of the various subperiods inthe cycle can be related to each other in the following way: (cid:102) LB ( V ib ) ( z ) = (cid:102) LB ( V ia ) (cid:0) q ( V i )1 z + p ( V i )1 , . . . , q ( V i ) N z N + p ( V i ) N (cid:1) , (3.1) (cid:102) LB ( S ia ) ( z ) = (cid:102) LB ( V ib ) (cid:0) z , . . . , z i − , h i ( z ) , z i +1 , . . . , z N (cid:1) , (3.2) (cid:102) LB ( S ib ) ( z ) = (cid:102) LB ( S ia ) (cid:0) q ( S i )1 z + p ( S i )1 , . . . , q ( S i ) N z N + p ( S i ) N (cid:1) , (3.3) (cid:102) LB ( V ( i +1) a ) ( z ) = (cid:102) LB ( S ib ) ( z ) σ i (cid:0) N (cid:88) j =1 λ j (1 − z j ) (cid:1) , (3.4)where we use the shorthand notation z for the vector ( z , . . . , z n ). Successive substitutionleads to a recursive expression for the joint queue length PGF at an arbitrary polling epoch.In, e.g., [22] it is discussed how this recursive expression leads to the PGF of the joint queuelength distribution at polling epochs, written as an infinite product. The recursive equationitself can be used to compute the moments of this joint queue length distribution explicitly.For now, we are more interested in a nice property of the Poisson arrival processes. During theactual visit period V ib , the type i customers that have not abandoned the system are served.For the moment, assuming exhaustive service, we focus on the end of V i , when there are notype i customers present in the system. Then the PGF of the number of type i customerspresent at the end of S i (which coincides with the beginning of V i +1 ) is (cid:102) LB ( V ( i +1) a ) (1 , . . . , , z i , , . . . ,
1) = σ i (cid:0) λ i (1 − z i ) (cid:1) . Each of these customers abandons the system before the start of V i +1 with probability p ( V i +1 ) i ,so the PGF of the number of type i customers that are still in the system at the beginningof V i +1 is: (cid:102) LB ( V ( i +1) b ) (1 , . . . , , z i , , . . . ,
1) = (cid:102) LB ( V ( i +1) a ) (1 , . . . , , q ( V i +1 ) i z i + p ( V i +1 ) i , , . . . , σ i (cid:0) λ i (1 − q ( V i +1 ) i z i − p ( V i +1 ) i ) (cid:1) = σ i (cid:0) λ i q ( V i +1 ) i (1 − z i ) (cid:1) . This short example illustrates that, as far as the joint queue lengths at polling epochs isconcerned, and only focussing on the customers that did not abandon the system prematurely,we can view the system as a polling model with Poisson arrivals, but with varying arrivalrates (in the example, we have that the new arrival rate is λ i q ( V i +1 ) i during S i ). Just before thestart of S i +1 each type i customer in the system reneges with probability p ( S i +1 ) i . This impliesthat a customer that arrived during S i is still in the system at the beginning of S i +1 withprobability q ( V i +1 ) i q ( S i +1 ) i . Hence, the number of type i customers present at the beginning of S i +1 is the same as in a polling system without reneging, but with arrival rates q ( V i +1 ) i q ( S i +1 ) i λ i during S i , and q ( S i +1 ) i λ i during V i +1 . This observation makes it possible to analyse the pollingsystem with reneging by regarding a dual system, without reneging but with varying arrivalrates. For the remainder of this section, we consider this dual system. A system with arrivalrates that depend on the location of the server, is studied in [5], where it is referred to as a6olling system with smart customers. We apply their results and adopt their notation. Let λ ( P ) i denote the arrival intensity of type i customers during period P ∈ { V , S , . . . , V N , S N } .In order to create a similar system as the original polling system with reneging, we definethese arrival intensities in the following way: λ ( S i − ) i = λ i q ( V i ) i ,λ ( V i − ) i = λ i q ( S i − ) i q ( V i ) i ,λ ( S i − ) i = λ i q ( V i − ) i q ( S i − ) i q ( V i ) i , ... (3.5) λ ( S i − N ) i = λ i q ( V i ) i N − (cid:89) j =1 q ( V i − j ) i q ( S i − j ) i ,λ ( V i − N ) i = (cid:40) λ i (cid:81) Nj =1 q ( V i − j ) i q ( S i − j ) i , if Q i receives gated service, λ i , if Q i receives exhaustive service.The only difference for gated service, compared to exhaustive service, is that type i customersarriving during V i have to wait until the next visit period of type i customers before they areserved. Thus, type i customers that arrive during V i , abandon the system before the startof S i with probability p ( S i ) i , whereas for exhaustive service all of these customers would beserved during the visit period in which they arrive. Cycle time
The cycle time distribution of this dual system is the same as in the original system withreneging. Theorem 5 . C and the intervisit time I : E (cid:2) e − ωC (cid:3) = (cid:102) LB ( V b ) (cid:16) θ ( ψ ( V ) ( ω )) , . . . , θ N ( ψ ( V N ) ( ω )) (cid:17) N (cid:89) i =1 σ i (cid:16) ψ ( S i ) ( ω ) (cid:17) , E (cid:2) e − ωI (cid:3) = (cid:102) LB ( S b ) (cid:16) , θ ( ψ ( V ) ( ω )) , . . . , θ N ( ψ ( V N ) ( ω )) (cid:17) N (cid:89) i =1 σ i (cid:16) ψ ( S i ) ( ω ) (cid:17) , where the functions ψ ( P ) ( ω ) are defined in the following, recursive way: ψ ( V N ) ( ω ) = ω,ψ ( V i ) ( ω ) = ω + N (cid:88) k = i +1 λ ( V i ) k (cid:16) − θ k ( ψ ( V k ) ( ω )) (cid:17) , i = N − , . . . , ,ψ ( S N ) ( ω ) = ω,ψ ( S i ) ( ω ) = ω + N (cid:88) k = i +1 λ ( S i ) k (cid:16) − θ k ( ψ ( V k ) ( ω )) (cid:17) , i = N − , . . . , . emark 3.1 Specifically for exhaustive and gated service, more compact expressions for theLSTs of the cycle time and intervisit time distributions are found in Theorem 5 . i ( V j ) customer is atype i customer that arrives during V j . Using the analysis based on customer subtypes, wecan express the LSTs of the cycle time and intervisit time in terms of the PGF of the jointqueue length distribution at polling instants of all customer subtypes, which we denote hereas (cid:102) LB ( V i ) (cid:0) z ( V )1 , . . . , z ( S N )1 , . . . , z ( V ) N , . . . , z ( S N ) N (cid:1) . We do not repeat the complete analysis onhow to obtain this PGF, but instead refer to Section 5 of [5].For exhaustive service, the LSTs of the distributions of C ∗ i and I i are: E (cid:2) e − ωC ∗ i (cid:3) = (cid:102) LB ( V i ) (cid:0) , . . . , , π i ( ω ) − ωλ ( V ) i , . . . , π i ( ω ) − ωλ ( S N ) i , , . . . , (cid:1) , (3.6) E (cid:2) e − ωI i (cid:3) = (cid:102) LB ( V i ) (cid:0) , . . . , , − ωλ ( V ) i , . . . , − ωλ ( S N ) i , , . . . , (cid:1) . (3.7)If Q i receives gated service , the LST of the cycle time distribution C i , and the LST of theintervisit time distribution I i , are given by: E (cid:2) e − ωC i (cid:3) = (cid:102) LB ( V i ) (cid:0) , . . . , , − ωλ ( V ) i , . . . , − ωλ ( S N ) i , , . . . , (cid:1) , E (cid:2) e − ωI i (cid:3) = (cid:102) LB ( V i ) (cid:0) , . . . , , , − ωλ ( S ) i , . . . , − ωλ ( S N ) i , , . . . , (cid:1) . Visit time
The LSTs of the distributions of the visit times V i , i = 1 , . . . , N , can directly be determinedfor any branching-type service discipline using the function θ i ( · ), and the joint queue lengthdistribution (without subtypes) at the beginning of subperiod V ib : E [e − ωV i ] = (cid:102) LB ( V ib ) (1 , . . . , , θ i ( ω ) , , . . . , . (3.8)The mean cycle time E [ C ] and mean visit times E [ V i ], which are needed later in this paper,can be obtained by differentiating the corresponding LSTs. A numerically more efficient wayto compute them, is using MVA for polling systems with smart customers, which is describedin more detail in [5]. Waiting time
The waiting time of customers in the dual system also has the same distribution as the timethat customers in the original system have to wait before being taken into service (not takingthe impatient customers into account). Note that the marginal queue length distribution at departure epochs of customers that did not renege, is not the same as the marginal queuelength distribution at arbitrary epochs , because the arrival intensities change during the cy-cle. This implies that we cannot use PASTA, and the standard distributional form of Little’slaw, as discussed by, e.g., Keilson and Servi [16], cannot be used to obtain the waiting time8istribution from the queue length distribution. What we can use though, is a slightly gener-alised version of the distributional form of Little’s law, that can be applied to the joint queuelength distribution at departure epochs, as discussed in the proof of Theorem 4 . i customer subtypes at a departure epoch from Q i , E (cid:34)(cid:16) z ( V ) i (cid:17) D ( V i · · · (cid:16) z ( S N ) i (cid:17) D ( SN ) i (cid:35) . Here, D ( P ) i is the number of type i ( P ) customers leftbehind at a departure epoch from Q i .The LST of the distribution of the waiting time W i of a type i customer, i = 1 , . . . , N , is: E (cid:2) e − ωW i (cid:3) = 1 β i ( ω ) E (cid:32) − ωλ ( V ) i (cid:33) D ( V i · · · (cid:32) − ωλ ( S N ) i (cid:33) D ( SN ) i . (3.9)In the next section we study the queue lengths of all customers that enter the system, includingthose that renege before the start of their service. Stability condition
The stability condition of a polling model with reneging at polling instants, is the same as inthe model with smart customers, discussed in the present section. It is clear that customersleaving the system without being served, do not contribute to the workload of the server.The stability condition of a model with smart customers is discussed in [5, 9]. In [9] it isshown that a necessary and sufficient condition for ergodicity is that the Perron-Frobeniuseigenvalue of the matrix R − I N should be less than 0, where I N is the N × N identity matrix,and R is an N × N matrix containing elements ρ ij := λ ( V j ) i E [ B i ]. Proportion of customers served
In queueing models with reneging, the expected proportion of customers that do not abandonthe system prematurely, denoted by r , is an important quality measure of the system. Insome systems, this might be difficult to compute. In the model considered in the presentpaper, this quantity is relatively easily obtained. The probability that an arbitrary customeris of type i , is obviously λ i / Λ, where Λ = (cid:80) Nj =1 λ j . The fraction of customers arriving duringperiod P ∈ { V , S , . . . , V N , S N } is E [ P ] / E [ C ]. Conditioning on the type of an arbitraryarriving customer, and the location of the server upon his arrival, one can determine theprobability that he will not abandon the system prematurely similarly to determining thearrival intensities (3.5). Denote by r i the probability that an arbitrary type i customer iseventually served. Then it is easily seen that r i = N (cid:88) j =1 (cid:32) E [ V j ] E [ C ] λ ( V j ) i λ i + E [ S j ] E [ C ] λ ( S j ) i λ i (cid:33) , i = 1 , . . . , N,r = N (cid:88) i =1 λ i Λ r i . Queue length distributions
In the previous section, we divided each visit period and switch-over period into two subpe-riods, part a where impatient customers decide to abandon the system, and part b , wherethe server is actually serving (or switching in the case of a switch-over period). The PGFsof the joint queue length distributions at the beginnings of all these subperiods are givenimplicitly by (3.1)–(3.4). In the present section we show how the marginal queue lengthdistribution at an arbitrary epoch can be expressed in terms of these PGFs. We denotethe marginal queue length of a type i customer by L i , i = 1 , . . . , N . The PGF of thedistribution of L i is determined by conditioning on the subperiod during which the queueis observed. The number of type i customers at an arbitrary moment in subperiod P ∈{ V a , V b , S a , S b , . . . , V Na , V Nb , S Na , S Nb } is denoted by L ( P ) i . By conditioning on P , wehave E [ z L i ] = N (cid:88) j =1 (cid:18) E [ V j ] E [ C ] E (cid:20) z L ( Vjb ) i (cid:21) + E [ S j ] E [ C ] E (cid:20) z L ( Sjb ) i (cid:21)(cid:19) , i = 1 , . . . , N, (4.1)where we used that E [ V ja ] = E [ S ja ] = 0, E [ V jb ] = E [ V j ], and E [ S jb ] = E [ S j ].Since S j , j = 1 , . . . , N and V j , j (cid:54) = i , are non-serving intervals for customers of type i , we usea standard result (see, e.g., [6]) to find the PGFs of L ( V jb ) i and L ( S jb ) i respectively: E (cid:20) z L ( Vjb ) i (cid:21) = E (cid:20) z LB ( Vjb ) i (cid:21) − E (cid:20) z LB ( Sja ) i (cid:21) (1 − z ) (cid:16) E [ LB ( S ja ) i ] − E [ LB ( V jb ) i ] (cid:17) , i = 1 , . . . , N ; j (cid:54) = i, (4.2) E (cid:20) z L ( Sjb ) i (cid:21) = E (cid:20) z LB ( Sjb ) i (cid:21) − E (cid:20) z LB ( V ( j +1) a ) i (cid:21) (1 − z ) (cid:16) E [ LB ( V ( j +1) a ) i ] − E [ LB ( S jb ) i ] (cid:17) , i, j = 1 , . . . , N, (4.3)where LB ( V jb ) i and LB ( S ja ) i are the number of type i customers at respectively a visit be-ginning and completion at Q j . Their PGFs are given by (cid:102) LB ( V jb ) (1 , . . . , , z, , . . . ,
1) and (cid:102) LB ( S ja ) (1 , . . . , , z, , . . . , z is the element at position i . Differentiation of thesePGFs and substituting z = 1 gives the mean values. Similarly, LB ( S jb ) i and LB ( V ( j +1) a ) i arethe number of type i customers at respectively the beginning and ending of S j .It remains to compute E (cid:104) z L ( Vi ) i (cid:105) , i = 1 , . . . , N , i.e. the PGF of the number of type i customersat an arbitrary epoch within V i . In order to do this, we temporarily look at a polling system without reneging, and focus on type i customers. As far as the marginal queue length of type i customers is concerned, the system can be viewed as a vacation queue where the intervisittime I i corresponds to the server vacation. In this “ordinary” polling model, we can use theFuhrmann-Cooper decomposition [13], which states that E [ z L i ] = (1 − λ i E [ B i ])(1 − z ) β i (cid:0) λ i (1 − z ) (cid:1) β i (cid:0) λ i (1 − z ) (cid:1) − z × E (cid:104) z LB ( Si ) i (cid:105) − E (cid:104) z LB ( Vi ) i (cid:105) (1 − z ) (cid:16) E [ LB ( V i ) i ] − E [ LB ( S i ) i ] (cid:17) , (4.4)where LB ( V i ) i and LB ( S i ) i denote the number of type i customers at respectively the beginningand completion of V i . The first part in this decomposition is the PGF of the marginal queue10ength of an M/G/ i customers only. The second part, which is independentof the first, is the PGF of the number of type i customers at an arbitrary epoch during theintervisit time I i , which we denote by E [ z L ( Ii ) i ]. Now we focus on the visit and intervisittime of Q i separately, using the relation E [ z L i ] = E [ V i ] E [ C ] E [ z L ( Vi ) i ] + E [ I i ] E [ C ] E [ z L ( Ii ) i ]. Plugging thisrelation into (4.4), leads to: E [ z L ( Vi ) i ] = 1 − λ i E [ B i ] λ i E [ B i ] z (cid:0) − β i ( λ i (1 − z )) (cid:1) β i ( λ i (1 − z )) − z × E (cid:104) z LB ( Si ) i (cid:105) − E (cid:104) z LB ( Vi ) i (cid:105) (1 − z ) (cid:16) E [ LB ( V i ) i ] − E [ LB ( S i ) i ] (cid:17) . (4.5)The second part of this decomposition is, again, the PGF of the number of customers at anarbitrary moment during the intervisit time I i . The first part can be recognised as the PGFof the queue length of an M/G/ i customers only, at an arbitrary epoch during a busy period.Now we return to the model with synchronised reneging. The key observation is that duringa visit period , this system behaves exactly as a polling system without reneging. Equation(4.5) no longer depends on anything that happens during the intervisit time, because thisis all captured in LB ( V i ) i , the number of type i customers at the beginning of a visit to Q i .This implies that (4.5) also holds for the system considered in the present paper. The onlydifference is that the interpretation of (4.5) is different. Obviously, the first part in (4.5) stillis the PGF of the queue length of an M/G/ i customers at an arbitrary moment during the intervisit time I i .For this reason, the Fuhrmann-Cooper decomposition does not hold in a polling model withsynchronised reneging. The condition that has to be satisfied for the Fuhrmann-Cooperdecomposition, is: (cid:88) j (cid:54) = i (cid:18) E [ V j ] E [ C ] E (cid:20) z L ( Vjb ) i (cid:21)(cid:19) + N (cid:88) j =1 (cid:18) E [ S j ] E [ C ] E (cid:20) z L ( Sjb ) i (cid:21)(cid:19) = E [ I i ] E [ C ] E (cid:104) z LB ( Si ) i (cid:105) − E (cid:104) z LB ( Vi ) i (cid:105) (1 − z ) (cid:16) E [ LB ( V i ) i ] − E [ LB ( S i ) i ] (cid:17) , (4.6)the left-hand side of (4.6) being E [ I i ] E [ C ] E [ z L ( Ii ) i ]. However, substitution of (4.2) and (4.3) in (4.6),and using (3.5), shows that (4.6) is only true if q ( P ) i = 1 for all P ∈ { V , S , . . . , V N , S N } ,because only in that case all terms in the numerator, except for E (cid:104) z LB ( Si ) i (cid:105) and E (cid:104) z LB ( Vi ) i (cid:105) cancel out.Substitution of (4.2), (4.3), and (4.5) in (4.1) gives the desired expression for the PGF of themarginal queue length in Q i . Additional remarks
The marginal queue length distribution at an arbitrary epoch is given by (4.1). It is notewor-thy that, unlike in Section 3, we can use PASTA now because we focus on all customers - in-cluding the impatient ones. This implies that the marginal queue length distribution at arrivaland departure epochs of type i customers is also given by (4.1). It should be noted that, whenstudying the queue length at departure epochs, we assume that reneging customers leave the11ystem in order of arrival, even though several of them might leave at the same renegingepoch. When looking at it this way, we do not really have group departures, but consecutivedepartures that might take place during an interval of zero length. Determining the LST ofthe sojourn time distribution of type i customers, including those that abandon the systembefore being served, remains difficult. One cannot use the distributional form of Little’s law,because there are multiple occasions within a cycle during which a type i customer mightleave the system. In the present paper we do not show exactly how to compute the sojourntime distribution of an arbitrary customer, because it requires a lot of bookkeeping. Foreach customer type, one needs to keep track of when a customer entered the system, and atwhich point he is going to leave the system. There are N customer types, entering the systemduring 2 N subperiods, and leaving the system at 2 N + 1 occasions (2 N reneging momentsplus one visit period). This gives a maximum of 2(2 N + 1) N customer subtypes, althoughit is determined by the service disciplines in the different queues, which of these subtypes areactually needed. For all customer subtypes, the joint queue lengths at departure momentshave to be determined in order to find the sojourn time distributions. We only show how thisis done for a vacation model, in Section 5. Although not applicable in its distributional form,Little’s law can still be used to determine the mean sojourn time of a type i customer, T i , for i = 1 , . . . , N : E [ T i ] = E [ L i ] /λ i . This section discusses the special case N = 1 and exhaustive service in more detail. Theresults obtained in the previous sections reduce to nice, compact expressions. When N = 1,there is only one queue being served, and the switch-over time between successive visits iscalled a server vacation . This model is studied in Adan et al. [1], where it is referred to as theUnique Abandonment Epoch (UAE) model. In [1] only the queue length PGF is determined.The methods used in the present paper also make it possible to find the LSTs of the cycletime and the waiting time distribution. Although it is not discussed explicitly in this section,of course it is also possible to analyse the vacation system with gated service.We use the same notation as in the rest of the paper, which is slightly different from commonnotation in vacation models. Since there is only one queue, the indices i = 1 , . . . , N , aredropped. A vacation is the switch-over period S with LST σ ( · ), whereas V denotes the visitperiod. The analysis in the present section is a slightly more extended version of the one inthe previous sections, because we aim at finding the sojourn time of an arbitrary customer,including those that renege, as well. This requires distinguishing not only between moments atwhich customers abandon the system, but also between their arrival (sub)periods. Therefore,the cycle is divided into four subperiods: V a , V ( S ) b , V ( V ) b and S . During V a the impatientcustomers abandon the system. All of these customers have arrived during S . The remainingcustomers, that have also arrived during S , are served during V ( S ) b . During V ( V ) b , all customersthat have entered the system during V ( S ) b , and newly arriving customers, are served until thesystem is empty. Since the system is empty at the beginning of S , we do not split S intosubperiods, and we use the notation p for the probability that a customer abandons the systembefore being served, and q = 1 − p . The astute reader has noticed that we distinguish between three customer types. Type a customers are those that abandon the system during V a , type12 ( S ) customers are those that enter during the vacation S without abandoning the system,type b ( V ) customers enter during the visit period and are always served. The advantage ofconsidering both the arrival epoch and the departure epoch of each customer type, is that itenables us to combine the techniques from Sections 3 and 4. The PGFs of the joint queuelength distributions, at the start of the four subperiods, follow from (3.1)–(3.4), and the LSTof the cycle time C ∗ follows from (3.6). In this vacation model, it reduces to: γ ∗ ( ω ) = (cid:102) LB ( V ( S ) b ) (1 , π ( ω ) − ωqλ ,
1) = σ (cid:0) ω + qλ (1 − π ( ω )) (cid:1) . The mean cycle time and the mean visit time are: E [ C ] = 1 − pρ − ρ E [ S ] , E [ V ] = qρ − ρ E [ S ] , where ρ = λ E [ B ]. In [1], the PGF of the marginal queue length distribution is obtainedusing similar techniques as in Section 4. In this section we use a different approach, basedon the joint queue length distribution at departure epochs. This approach is similar to theone used in Section 3, but now including the customers that renege from the system. Wefollow the steps taken by Borst [6], who extends an idea of Eisenberg [11], to find the PGFof the joint distribution of the queue lengths and state of the server at departure epochs, M ( P ) ( z a , z ( S ) b , z ( V ) b ). The state of the server is identified by the subperiod P , which can beany of the three periods during which customers depart from the system, V a , V ( S ) b and V ( V ) b . M ( V a ) ( z a , z ( S ) b , z ( V ) b ) = 1 λ E [ C ] 1 z a − (cid:18) (cid:102) LB ( V a ) ( z a , z ( S ) b , z ( V ) b ) − (cid:102) LB ( V ( S ) b ) ( z a , z ( S ) b , z ( V ) b ) (cid:19) ,M ( V ( S ) b ) ( z a , z ( S ) b , z ( V ) b ) = 1 λ E [ C ] β (cid:0) λ (1 − z ( V ) b ) (cid:1) z ( S ) b − β (cid:0) λ (1 − z ( V ) b ) (cid:1) × (cid:18) (cid:102) LB ( V ( S ) b ) ( z a , z ( S ) b , z ( V ) b ) − (cid:102) LB ( V ( V ) b ) ( z a , z ( S ) b , z ( V ) b ) (cid:19) ,M ( V ( V ) b ) ( z a , z ( S ) b , z ( V ) b ) = 1 λ E [ C ] β (cid:0) λ (1 − z ( V ) b ) (cid:1) z ( V ) b − β (cid:0) λ (1 − z ( V ) b ) (cid:1) (cid:18) (cid:102) LB ( V ( V ) b ) ( z a , z ( S ) b , z ( V ) b ) − (cid:19) . The PGF of the joint queue length distribution at an arbitrary departure epoch is simplythe sum of these three PGFs. Using PASTA and an up-and-down crossing argument, we findthat the marginal queue length distribution at an arbitrary moment is: E [ z L ] = M ( V a ) ( z, z, z ) + M ( V ( S ) b ) ( z, z, z ) + M ( V ( V ) b ) ( z, z, z ) . The sojourn time of an arbitrary customer in a polling model with synchronised reneginghas not been discussed in the present paper because of the effort it takes to keep track of allcustomers and their arrival and departure moments. For this vacation model, it is not toocomplicated though. It requires applying the generalised distributional form of Little’s law,as discussed in [5], to the joint queue length distribution at departure epochs. This leads to13he following LST of the distribution of the sojourn time T of an arbitrary customer: E [e − ωT ] = M ( V a ) (cid:0) − ωpλ , , (cid:1) + M ( V ( S ) b ) (cid:0) , − ωqλ , − ωλ (cid:1) + M ( V ( V ) b ) (cid:0) , − ωqλ , − ωλ (cid:1) = p (1 − ρ )1 − pρ − σ ( ω ) ω E [ S ] + q (1 − ρ )1 − pρ σ (cid:0) ω (cid:1) − σ (cid:0) qλ (1 − β ( ω )) (cid:1)(cid:0) qλ (1 − β ( ω )) − ω (cid:1) E [ S ] β ( ω )+ 1 − ρ − pρ σ (cid:0) qλ (1 − β ( ω )) (cid:1) − (cid:0) λ (1 − β ( ω )) − ω (cid:1) E [ S ] β ( ω ) . (5.1)The waiting time of customers that did not renege the system, is obtained in the same way,but without taking into account the type a customers: E [e − ωW b ] = λ E [ C ] qλ E [ S ] + λ E [ V ] (cid:18) M ( V ( S ) b ) (cid:0) , − ωqλ , − ωλ (cid:1) + M ( V ( V ) b ) (cid:0) , − ωqλ , − ωλ (cid:1)(cid:19) β ( ω ) . In this section we study the impact of the reneging probabilities on the mean queue lengthsin a two-queue polling system. The service times of all customers, in both queues, are ex-ponentially distributed with mean 1. The arrival processes are Poisson with rates for Q and for Q . We deliberately choose an imbalanced system to study differences between aheavily and a lightly loaded queue. The switch-over times are also exponentially distributed.We compare a system with small switch-over times, E [ S i ] = 1, with a system having largerswitch-over times, E [ S i ] = 10. Finally, two different combinations of reneging probabilitiesare taken:Case 1: p ( V j ) i = p ( S j ) i = p i , (6.1)Case 2: p ( S i ) i = 810 p i , p ( V i +1 ) i = 610 p i , p ( S i +1 ) i = 410 p i , p ( V i ) i = 210 p i , (6.2)for i, j = 1 ,
2. In Case 1, the reneging probabilities per customer type are the same for allreneging moments. In Case 2, which might be considered as more realistic, the probabilitiesof reneging decrease as the moment of being served comes nearer. The parameters p i arevaried, independently for i = 1 ,
2, between 0 and 1. Furthermore, we study all possiblecombinations of gated and exhaustive service for each queue. The results are depicted inFigures 1 – 4, where the mean queue lengths E [ L ] and E [ L ] are plotted against p and p .As expected, Q dominates the behaviour of the system, because of its heavy load comparedto Q . For this reason, results are omitted for Q receiving exhaustive service, because theyhardly deviate from the results where Q receives gated service. A conclusion that can bedrawn from a comparison between Figures 2 and 4, is that the lengths of the switch-overtimes hardly influence the impact of p and p on the mean queue lengths if the renegingprobabilities are decreasing as in Case 2. For constant reneging probabilities, as in Case 1,the behaviour of the mean queue lengths changes when the mean switch-over times becomelarger. The non-monotonic behaviour that was noted in one of the examples studied in [1],is also visible in Figure 3. If switch-over times are relatively large, higher values of p mayresult in an increase in the mean number of customers in Q , but also in Q if Q receives14ated service. Furthermore, it is interesting to observe in Figures 2 and 4 that in Case 2,both p and p have a high impact on E [ L ] and E [ L ]. In contrast, for Case 1, the influenceof p and p varies per queue. E.g., Figures 1 and 3 illustrate that E [ L ] is mainly influencedby p , whereas E [ L ] is influenced by both parameters.Case 1: constant reneging probabilities, E [ S i ] = 1 Q gated Q exhaustive p p p p Q gated Q gated p p p p Figure 1: Mean queue lengths in the polling system in Case 1 of Example 2, versus p and p . The reneging probabilities are constant, E [ S i ] = 1. In the present section we summarise our findings and conclusions, and discuss possible futureextensions of the model under consideration. We have extended some results on a vacationmodel with synchronised reneging, as presented in [1], to a polling system consisting of mul-tiple queues. Using techniques from a polling model with varying arrival rates, depending onthe server location (cf. [5]) we have been able to find the LST of the cycle time distribution,visit and intervisit time distributions, and of the waiting time distribution of customers thatget served eventually. An adaptation of these techniques leads to the PGF of the marginalqueue length distribution of all customers in each queue. It also leads to the sojourn time dis-tribution, but this requires lengthy, cumbersome computations that have only been discussedfor a system consisting of one queue.When comparing the model of the present paper with existing literature, one of the moststriking differences is the non-monotonic behaviour of the mean queue lengths that mightoccur in polling (or vacation) models. As illustrated in the numerical example, the pres-ence of large switch-over times might cause the mean queue lengths to increase if renegingprobabilities increase. Another notable difference is that it is relatively easy to compute the15ase 2: decreasing reneging probabilities, E [ S i ] = 1 Q gated Q exhaustive p p p p Q gated Q gated p p p p Figure 2: Mean queue lengths in the polling system in Case 2 of Example 2, versus p and p . The reneging probabilities are decreasing, E [ S i ] = 1.Case 1: constant reneging probabilities, E [ S i ] = 10 Q gated Q exhaustive p p p p Q gated Q gated p p p p Figure 3: Mean queue lengths in the polling system in Case 1 of Example 2, versus p and p . The reneging probabilities are constant, E [ S i ] = 10.16ase 2: decreasing reneging probabilities, E [ S i ] = 10 Q gated Q exhaustive p p p p Q gated Q gated p p p p Figure 4: Mean queue lengths in the polling system in Case 2 of Example 2, versus p and p . The reneging probabilities are decreasing, E [ S i ] = 10.proportion of customers served. This metric is a relevant quantity in reneging literature. Inour model, given the customer type and server location upon his arrival, one knows exactlyhow many possibilities this customer will have to abandon the system, and the correspondingreneging probabilities.Several research topics, beyond the scope of the present paper, are worth studying. Aninteresting extension is to allow (synchronised) reneging at various epochs during switch-overand visit periods. Possibly, the analysis of the Multiple Abandonment Epochs (MAE) modelin [1] (synchronised reneging), and the analysis of Altman and Yechiali [2] (customers growingimpatient during vacations) might be extended to polling models. Another extension, relevantfrom a practical point of view, is to develop numerically more efficient algorithms to computeperformance measures of interest. In the present paper, we use the buffer occupancy method,but it would be more efficient to extend the Mean Value Analysis (MVA) framework for pollingsystems (cf. [27]) to a polling model with reneging at polling instants. In [1], as well as in [5],MVA has been used to find the mean queue lengths. These two implementations should givea good indication of how to implement MVA for the model discussed in the present paper.Note that difficulties in finding the sojourn time distribution do not occur when studying the mean sojourn time, which can simply be found using Little’s law. Acknowledgements
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