Waiting times in queueing networks with a single shared server
WWaiting times in queueing networks with a single shared server ∗ M.A.A. Boon † [email protected] R.D. van der Mei ‡ [email protected] E.M.M. Winands § [email protected], 2012 Abstract
We study a queueing network with a single shared server that serves the queues in acyclic order. External customers arrive at the queues according to independent Poissonprocesses. After completing service, a customer either leaves the system or is routedto another queue. This model is very generic and finds many applications in computersystems, communication networks, manufacturing systems, and robotics. Special cases ofthe introduced network include well-known polling models, tandem queues, systems with awaiting room, multi-stage models with parallel queues, and many others. A complicatingfactor of this model is that the internally rerouted customers do not arrive at the variousqueues according to a Poisson process, causing standard techniques to find waiting-timedistributions to fail. In this paper we develop a new method to obtain exact expressionsfor the Laplace-Stieltjes transforms of the steady-state waiting-time distributions. Thismethod can be applied to a wide variety of models which lacked an analysis of the waiting-time distribution until now.
Keywords: queueing network, waiting times, customer routing, shared server, polling
Mathematics Subject Classification:
In this paper we study a queueing network served by a single shared server that visits thequeues in a cyclic order. Customers from the outside arrive at the queues according toindependent Poisson processes, and the service time and switch-over time distributions aregeneral. After receiving service at queue i , a customer is either routed to queue j with ∗ The research was done in the framework of the BSIK/BRICKS project, the European Network of ExcellenceEuro-NF, and of the project “Service Optimization and Quality” (SeQual), funded by the Dutch agencySenterNovem. † Eurandom and Department of Mathematics and Computer Science, Eindhoven University of Technology,P.O. Box 513, 5600MB Eindhoven, The Netherlands ‡ Department of Mathematics, Section Stochastics, VU University, De Boelelaan 1081a, 1081HV Amster-dam, The Netherlands and Centre for Mathematics and Computer Science (CWI), 1098 SJ Amsterdam, TheNetherlands § University of Amsterdam, Korteweg-de Vries Institute for Mathematics, Science Park 904, 1098 XH Am-sterdam, The Netherlands a r X i v : . [ m a t h . P R ] A ug robability p i,j , or leaves the system with probability p i, . We consider systems with mixturesof gated and exhaustive service. This model can be seen as an extension of the standardpolling model (in which customers always leave the system upon completion of their service)by customer routing. Yet another view is provided by the notion that the system is a Jacksonnetwork with a dedicated server for each queue with the additional complexity that only oneserver can be active in the network simultaneously.The possibility of re-routing of customers further enhances the already-extensive modellingcapabilities of polling models, which find applications in diverse areas such as computersystems, communication networks, logistics, flexible manufacturing systems, robotics systems,production systems and maintenance systems (see, for example, [5, 18, 22, 32] for overviews).Applications of the introduced type of customer routing can be found in many of these areas.In this regard, we would like to mention a manufacturing system where products undergoservice in a number of stages or in the context of rework [17], a Ferry based Wireless LocalArea Network (FWLAN) in which nodes can communicate with each other or with the outerworld via a message ferry [20], a dynamic order picking system where the order picker dropsoff the picked items at the depot where sorting of the items is performed [16], and an internalmail delivery system where a clerk continuously makes rounds within the offices to pick up,sort and deliver mail [27].In the past many papers have been published on special cases of the current network. In someof these papers distributional results are derived as well; the techniques used do, however, notallow for extension to the general setting of the current paper. Some special case configurationsare standard polling systems [32], tandem queues [23, 34], multi-stage queueing models withparallel queues [19], feedback vacation queues [9, 33], symmetric feedback polling systems[31, 33], systems with a waiting room [1, 30], and many others. In conclusion, one can saythat the present research can be seen as a unifying analysis of the waiting-time distributionfor a wide variety of queueing models.The main contribution of this paper is the derivation of waiting-time distributions in queueingnetworks with a single roving server via the development of a new method. For this modelwe derive the Laplace-Stieltjes transform of the waiting-time distribution of an arbitrary(internally rerouted, or external) customer. Due to this intrinsic complexity of the model,studies in the past were restricted to queue lengths and mean delay figures (see [6, 27, 28, 29]).A complicating, yet interesting, factor is that the combined process of internal and externalarrivals violates the classical assumption of Poisson (or even renewal) arrivals, implying thattraditional methods are not applicable. The basic idea behind the new method is that weexplicitly compute a priori all future service requirements upon arrival of a newly arrivingcustomer. In doing so the prerequisites of the distributional form of Little’s Law are overcome.An important feature of the newly developed technique is that it can be applied to a myriad ofmodels which lacked an analysis of the waiting-time distribution until now. One could applythe framework (possibly after some minor modifications) to obtain distributional results in allof the aforementioned special cases of the studied system [1, 9, 19, 23, 30, 31, 32, 33, 34] butalso, for example, in a closed network [2], in an M/G/
We consider a queueing network consisting of N ≥ Q , . . . , Q N .External customers arrive at Q i according to a Poisson arrival process with rate λ i , and havea generally distributed service requirement B i at Q i , with mean value b i := E [ B i ]. In generalwe denote the Laplace-Stieltjes Transform (LST) or Probability Generating Function (PGF)of a random variable X with (cid:101) X ( · ). The queues are served by a single server in cyclic order.Whenever the server switches from Q i to Q i +1 , a random switch-over time R i is incurred, withmean r i . The cycle time C i is the time between successive moments when the server arrives at Q i . The total switch-over time in a cycle is denoted by R = (cid:80) Ni =1 R i , and its first two momentsare r := E [ R ] and r (2) := E [ R ]. Indices throughout the paper are modulo N , so Q − N and Q N +1 both refer to Q . All service times and switch-over times are mutually independent.This queueing network can be modelled as a polling system with the specific feature that itallows for routing of the customers: upon completion of service at Q i , a customer is eitherrouted to Q j with probability p i,j , or leaves the system with probability p i, . Note that p i, should be greater than 0 for at least one queue, to make sure that customers can leave thesystem eventually. Moreover, note that (cid:80) Nj =0 p i,j = 1 for all i , and that the transition of acustomer from Q i to Q j takes no time. Since we consider the gated and exhaustive servicedisciplines, the model under consideration has a branching structure, which is discussed inmore detail by Foss [15] in the context of queueing models, and by Resing [26] more specificallyin the context of polling systems. The total arrival rate at Q i is denoted by γ i , which is theunique solution of the following set of linear equations: γ i = λ i + N (cid:88) j =1 γ j p j,i , i = 1 , . . . , N. The offered load to Q i is ρ i := γ i b i and the total load is ρ := (cid:80) Ni =1 ρ i . We assume that thesystem is stable, which means that ρ should be less than one (see [29]). In the present section we study the waiting-time distribution of an arbitrary customer for asystem in which each queue receives gated service, which means that only those customerspresent at the server’s arrival at Q i will be served before the server switches to the nextqueue. We define the waiting time W i of an arbitrary customer in Q i as the time between hisarrival at this queue and the moment at which his service starts. As far as waiting times are3oncerned, a customer that is routed to another queue, say Q j , upon his service completion isregarded as a new customer with waiting time W j . The waiting-time distribution is found byconditioning on the numbers of customers present in each queue at an arrival epoch. To thisend, we study the joint queue-length distribution at several embedded epochs in Section 3.1.In Sections 3.2 and 3.3 we use these results to successively derive the cycle-time distributionand the waiting-time distributions of internally rerouted customers and external customers. Sidi et al. [29] derive the PGFs of the joint queue-length distributions in all N queues atvisit beginnings, visit completions, and at arbitrary points in time. In order to keep thismanuscript self-contained, we briefly recapitulate their approach, as it forms the startingpoint of our novel method to find the waiting time LSTs. There is one important adaptationthat we have to make, which will prove essential for finding waiting time LSTs. We considernot only the customers in all N queues, but we distinguish between customers standing infront of the gate and customers standing behind the gate (meaning that they will be served inthe next cycle). Hence, we introduce the N + 1 dimensional vector z = ( z , . . . , z N , z G ). Theelement z i , i = 1 , . . . , N , in this vector corresponds to customers in Q i standing in front of thegate. The element z G at position N + 1 is only used during visit periods. During V j , the visitperiod of Q j , it corresponds to customers standing behind the gate in Q j . This makes theanalysis of systems with gated service slightly more involved than systems with exhaustiveservice (discussed in the next section). Before studying the joint queue-length distributions,we briefly introduce some convenient notation:Σ( z ) = N (cid:88) j =1 λ j (1 − z j ) , Σ i ( z ) = λ i (1 − z G ) + (cid:88) j (cid:54) = i λ j (1 − z j ) ,P i ( z ) = p i, + p i,i z G + (cid:88) j (cid:54) = i p i,j z j . Visit beginnings and completions.
A cycle consists of N visit periods, V i , each of whichis followed by a switch-over time R i , for i = 1 , . . . , N . A cycle C i starts with a visit to Q i andconsists of the periods V i , R i , V i +1 , . . . , V i + N − , R i + N − . Let P denote any of these periods.We denote the joint queue length PGF at the beginning of P as (cid:102) LB ( P ) ( z ). The equivalentat the completion of period P is denoted by (cid:102) LC ( P ) ( z ). Since the gated service discipline is aso-called branching-type service discipline (see [26]), we can express each of these functions interms of (cid:102) LB ( V i ) ( z ), for any i = 1 , . . . , N . These relations, which are sometimes called laws of otion , are given below. (cid:102) LC ( V i ) ( z ) = (cid:102) LB ( V i ) (cid:16) z , . . . , z i − , (cid:101) B i (cid:0) Σ i ( z ) (cid:1) P i ( z ) , z i +1 , . . . , z N , z G (cid:17) , (3.1a) (cid:102) LB ( R i ) ( z ) = (cid:102) LC ( V i ) ( z , . . . , z N , z i ) , (3.1b) (cid:102) LC ( R i ) ( z ) = (cid:102) LB ( R i ) ( z ) (cid:101) R i (cid:16) Σ( z ) (cid:17) , (3.1c) (cid:102) LB ( V i +1 ) ( z ) = (cid:102) LC ( R i ) ( z ) , (3.1d)... (cid:102) LB ( V i + N ) ( z ) = (cid:102) LC ( R i + N − ) ( z ) . (3.1e)Note the subtle difference between (cid:102) LC ( V i ) ( z ) and (cid:102) LB ( R i ) ( z ), due to the fact that the gate in Q i is removed after the completion of V i , causing type G customers to become type i customers.In steady-state we have that (cid:102) LB ( V i + N ) ( z ) = (cid:102) LB ( V i ) ( z ), implying that we have obtained arecursive relation for (cid:102) LB ( V i ) ( z ). Resing [26] shows how a clever definition of immigration andoffspring generating functions can be used to find an explicit expression for (cid:102) LB ( V i ) ( z ). Forreasons of compactness we refrain from doing so in the present paper. Instead we want topoint out that the recursive relation obtained from (3.1a)-(3.1e) can be differentiated withrespect to the variables z , . . . , z N , z G . The resulting set of equations, which are called the buffer occupancy equations in the polling literature, can be used to compute the moments ofthe queue-length distributions at all visit beginnings and completions. Service beginnings and completions.
We denote the joint queue length PGF at service beginnings and completions in Q j by respectively (cid:102) LB ( B j ) ( z ) and (cid:102) LC ( B j ) ( z ). Since a customermay be routed to another queue upon his service completion, we define (cid:102) LC ( B j ) ( z ) as thePGF of the joint queue-length distribution right after the tagged customer in Q j has receivedservice (implying that he is no longer present in Q j ), but before the moment that he mayjoin another queue (even though these two epochs take place in a time span of length zero).Eisenberg [14] observed the following relation, albeit in a slightly different model: (cid:102) LB ( V i ) ( z ) + γ i E [ C ] (cid:102) LC ( B i ) ( z ) P i ( z ) = (cid:102) LC ( V i ) ( z ) + γ i E [ C ] (cid:102) LB ( B i ) ( z ) . (3.2)Equation (3.2) is based on the observation that each visit beginning coincides with either aservice beginning, or a visit completion (if no customer was present). Similarly, each visitcompletion coincides with either a visit beginning or a service completion. The long-run ratiobetween the number of visit beginnings/completions and service beginnings/completions in Q i is γ i E [ C ], with E [ C ] = E [ C i ] = r/ (1 − ρ ). The distribution of the cycle time is given inthe next subsection.Furthermore, Eisenberg observes the following simple relation between the joint queue-lengthdistribution at service beginnings and completions: (cid:102) LC ( B i ) ( z ) = (cid:102) LB ( B i ) ( z ) (cid:101) B i (cid:0) Σ i ( z ) (cid:1) /z i . (3.3)5ubstitution of (3.3) in (3.2) gives an equation which can be solved to express (cid:102) LB ( B i ) ( z ) in (cid:102) LB ( V i ) ( z ) and (cid:102) LC ( V i ) ( z ). Arbitrary moments.
The PGF of the joint queue-length distribution at arbitrary mo-ments, denoted by (cid:101) L ( z ), is found by conditioning on the period in the cycle during which thesystem is observed ( V , R , . . . , V N , R N ). (cid:101) L ( z ) = 1 E [ C ] N (cid:88) j =1 (cid:16) E [ V j ] (cid:101) L ( V j ) ( z ) + r j (cid:101) L ( R j ) ( z ) (cid:17) , (3.4)with E [ V j ] = ρ j E [ C ]. In (3.4) the functions (cid:101) L ( V j ) ( z ) and (cid:101) L ( R j ) ( z ) denote the PGFs of thejoint queue-length distributions at an arbitrary moment during V j and R j respectively: (cid:101) L ( V j ) ( z ) = (cid:102) LB ( B j ) ( z ) 1 − (cid:101) B j (cid:0) Σ j ( z ) (cid:1) b j Σ j ( z ) , (3.5) (cid:101) L ( R j ) ( z ) = (cid:102) LB ( R j ) ( z ) 1 − (cid:101) R j (cid:0) Σ( z ) (cid:1) r j Σ( z ) . (3.6)The interpretation of (3.5) and (3.6) is that the queue length vector at an arbitrary timepoint in V j or R j is the sum of those customers that were present at the beginning of thatservice/switch-over time, plus vector of the customers that have arrived during the elapsedpart of the service/switch-over time. For more details about the joint queue length andworkload distributions for general branching-type service disciplines (in the context of pollingsystems, but also applicable to our model) we refer to Boxma et al. [11]. In the remainder of this paper we present new results for the model introduced in Section 2.We start by analysing the distributions of the cycle times C i , i = 1 , . . . , N . The idea behindthe following analysis is to condition on the number of customers present in each queue atthe beginning of C i (and, hence, of V i ). The cycle will consist of the service of all of thesecustomers, plus all switch-over times R i , . . . , R i + N − , plus the services of all customers thatenter during these services and switch-over times and will be served before the next visitbeginning to Q i . The cycle time for polling systems without customer routing is discussed inBoxma et al. [10]. However, as it turns out, the analysis is severely complicated by the factthat customers may be routed to another queue and be served again (even multiple times)during the same cycle.From branching theory we adopt the term descendants of a certain (tagged) customer todenote all customers that arrive (in all queues) during the service of this tagged customer, plusthe customers arriving during their service times, and so on. If, upon his service completion,a customer is routed to another queue, we also consider him as his own descendant. We define B ∗ k,i , i = 1 , . . . , N ; k = 0 , . . . , N , as the service time of a type i − k (which is understood as N + i − k if i ≤ k ) customer at Q i − k , plus the service times of all of his descendants that willbe served before or during the next visit of the server to Q i . The special case B ∗ ,i is simply6he service time of a type i customer, i = 1 , . . . , N . A formal definition in terms of LSTs isgiven below: (cid:101) B ∗ k,i ( ω ) = (cid:101) B i − k (cid:16) ω + k − (cid:88) j =0 λ i − j (cid:0) − (cid:101) B ∗ j,i ( ω ) (cid:1)(cid:17) (cid:101) P ∗ k,i ( ω ) , k = 0 , , . . . , N ; i = 1 , . . . , N, (3.7)where (cid:101) P ∗ k,i ( ω ) = 1 − k − (cid:88) j =0 p i − k,i − j (cid:0) − (cid:101) B ∗ j,i ( ω ) (cid:1) , k = 0 , , . . . , N ; i = 1 , . . . , N. (3.8)For a type i − k customer, P ∗ k,i accounts for the service times of his descendants that arecaused by the fact that he may be routed to another queue upon his service completion.A similar function should be defined for the switch-over times: (cid:101) R ∗ k,i ( ω ) = (cid:101) R i − k (cid:16) ω + k − (cid:88) j =0 λ i − j (cid:0) − (cid:101) B ∗ j,i ( ω ) (cid:1)(cid:17) , k = 0 , , . . . , N ; i = 1 , . . . , N. (3.9)Note that, compared to (3.7), no term (cid:101) P ∗ k,i ( ω ) is required because no routing takes place atthe end of a switch-over time.Finally, we define the following N + 1 dimensional vectors: B k , i = (cid:0) , . . . , , (cid:101) B ∗ k,i ( ω ) , , . . . , (cid:1) , k = 0 , , . . . , N − i = 1 , . . . , N, (3.10) B N , i = (cid:0) , . . . , , (cid:101) B ∗ ,i ( ω ) (cid:1) , i = 1 , . . . , N, (3.11)with (cid:101) B ∗ k,i ( ω ) at position i − k in (3.10) (or position N + i − k if k ≥ i ), and (cid:101) B ∗ ,i ( ω ) at position N + 1 in (3.11). We use (cid:78) to denote the element-wise multiplication of vectors. Proposition 3.1
The LST of the distribution of the cycle time C i is given by (cid:101) C i ( ω ) = (cid:102) LB ( V i ) (cid:0) N − (cid:79) k =0 B k , i − (cid:1) N − (cid:89) k =0 (cid:101) R ∗ k,i − ( ω ) , i = 1 , . . . , N. (3.12)The interpretation of (3.12) is that the length of a cycle starting with a visit to Q i is the sumof the extended service times of all customers present at the beginning of the cycle, and thesum of all extended switch-over times during the cycle. By extended service time (switch-overtime) we refer to a service time (switch-over time) plus the service times of all customersthat arrive during this service time (switch-over time) in one of the queues that are yet tobe served during the remainder of the cycle, and all of their descendants that will be servedbefore the end of the cycle. Proof:
To prove Proposition 3.1 we keep track of all the customers that will be servedduring one cycle. We condition on the numbers of customers present in each queue at thebeginning of C i , denoted by n , . . . , n N . Note that there are no gated customers present atthis moment, because the gate has been removed at the beginning of the last switch-over timeof the previous cycle. A cycle C i consists of: 7. the service of all customers present at the beginning of the cycle,2. all of their descendants that will be served before the start of the next cycle (i.e., beforethe next visit to Q i ),3. the switch-over times R , . . . , R N ,4. all customers arriving during these switch-over times that will be served before the startof the next cycle,5. all of their descendants that will be served before the start of the next cycle.We define S j for j = 1 , . . . , N , as the service time of a type j customer plus the service timesof all of his descendants that will be served during (the remaining part of) C i . Since theservice discipline is gated at all queues, we have: S j = B j + i − (cid:88) k = j +1 N k ( B j ) (cid:88) l =1 S k l + (cid:40) S m for m = j + 1 , . . . , i −
1, w.p. p j,m , − (cid:80) i − m = j +1 p j,m , (3.13)where N k ( T ) denotes the number of arrivals in Q k during a (possibly random) period of time T , and S k l is a sequence of (independent) extended service times S k . Note that S j dependson i , although we have chosen to hide this for presentational purposes. The gated servicediscipline is reflected in the fact that only customers arriving in (or rerouted to) Q j +1 , . . . , Q i − are being served during the residual part of C i . It can easily be shown that the LST of S i − k is (cid:101) B ∗ k − ,i − ( ω ) for k = 1 , . . . , N . Note that the first summation in (3.13) is cyclic, which maysometimes cause confusion (for example if j = i −
1, when this is supposed to be a summationover zero terms). Avoiding this (possible) confusion is the main reason that we have chosento define (cid:101) B ∗ k,i ( ω ), (cid:101) P ∗ k,i ( ω ) and (cid:101) R ∗ k,i ( ω ) relative to queue i ( k steps backward in time).Using this branching way of looking at the cycle time, we can express C i in terms of R , . . . , R N and S , . . . , S N . First, however, we derive the following intermediate result. E e − ωR i − k i − (cid:89) j = i − k +1 N j ( R j ) (cid:89) l =1 e − ωS jl = (cid:101) R i − k (cid:0) ω + i − (cid:88) j = i − k +1 λ j (1 − E [e − ωS j ]) (cid:1) = (cid:101) R ∗ k − ,i − ( ω ) . Now, introducing the shorthand notation n , . . . , n N for the event that the numbers of cus-tomers at the beginning of C i in queues 1 , . . . , N are respectively n , . . . , n N , we can find thecycle time LST conditional on this event. 8 (cid:2) e − ωC i | n , . . . , n N (cid:3) = E exp (cid:16) − ω i − (cid:88) j = i − N (cid:0) n j (cid:88) l =1 S j l + R j + i − (cid:88) k = j +1 N k ( R j ) (cid:88) l =1 S k l (cid:1)(cid:17) = E i − (cid:89) j = i − N (cid:32) n j (cid:89) l =1 e − ωS jl (cid:33) e − ωR j i − (cid:89) k = j +1 N k ( R j ) (cid:89) l =1 e − ωS kl = i − (cid:89) j = i − N (cid:32) n j (cid:89) l =1 E (cid:2) e − ωS jl (cid:3)(cid:33) i − (cid:89) j = i − N E e − ωR j i − (cid:89) k = j +1 N k ( R j ) (cid:89) l =1 (cid:16) e − ωS kl (cid:17) = (cid:32) N (cid:89) k =1 (cid:101) B ∗ k − ,i − ( ω ) n i − k (cid:33) N (cid:89) k =1 (cid:101) R ∗ k − ,i − ( ω ) . Equation (3.12) follows after deconditioning. (cid:3)
Remark 3.2
Because of our main interest in the waiting-time distributions, we have fol-lowed quite an elaborate path to find the LST of the cycle-time distribution. However, if oneis merely interested in a quick way to find (cid:101) C i ( ω ), a more efficient approach can be used. One ofthe most efficient ways to find (cid:101) C i ( ω ) is to distinguish between customers that arrive from out-side the network (external customers) and internally rerouted customers (internal customers).One can straightforwardly adapt the laws of motion (3.1a)-(3.1e) to find an expression for (cid:102) LB ( V i ) (cid:48) ( z E , z I , . . . , z EN , z IN ). Just like (cid:102) LB ( V i ) ( z , . . . , z N , z G ), (cid:102) LB ( V i ) (cid:48) ( z E , z I , . . . , z EN , z IN ) standsfor the PGF of the joint queue length at the beginning of V i , but now we distinguish be-tween external and internal customers in each queue (indicated by z Ej and z Ij ). Since externalcustomers arrive in Q i according to a Poisson process with intensity λ i , one can apply thedistributional form of Little’s Law (see, for example, Keilson and Servi [21]) to the external customers in Q i : (cid:101) C i ( ω ) = (cid:102) LB ( V i ) (cid:48) (1 , . . . , , − ω/λ i , , . . . , , i = 1 , . . . , N. In this subsection we find the LSTs of W Ei and W Ii , the waiting-time distributions of arbitraryexternal and internal customers in Q i , and use them to obtain the LST of W i , the waitingtime of an arbitrary customer. Recall that the waiting time W i of an arbitrary customerin Q i is the time between his arrival at this queue and the moment at which his servicestarts. Hence, even if a customer is routed to the same queue multiple times, each visit tothis queue invokes a new waiting time. We stress that common methods used in the pollingliterature to find waiting time LSTs cannot be applied in our queueing network, because theyrely heavily on the assumption that every customer in the system has arrived according to aPoisson process. Since this assumption is violated in our model, we have developed a novelapproach to find the waiting time LST of an arbitrary customer in our network. The jointqueue-length distributions at various epochs, as discussed in Subsection 3.1, play an essentialrole in the analysis. First we focus on the waiting times of internal customers, then we discussthe waiting times of external customers. 9 nternal customers. The arrival epoch of an internal customer always coincides with aservice completion. Hence, we condition on the joint queue length and the arrival epoch ofan internal customer to find his waiting time LST. The waiting time of an internal customer given that he arrives in Q i after a service completion at Q i − k is denoted by WC ( B i − k ) i ( i, k =1 , . . . , N ). To find WC ( B i − k ) i , we only have to compute the probability that an arbitraryinternal customer in Q i arrives after a service completion at Q i − k . The mean number ofcustomers (internal plus external) present at the beginning of V i − k at Q i − k is γ i − k E [ C ].Each of these customers joins Q i upon his service completion with probability p i − k,i . Thisobservation combined with the fact that the mean number of internal customers arriving at Q i during the course of one cycle is ( γ i − λ i ) E [ C ], leads to the following result: (cid:102) W Ii ( ω ) = N (cid:88) k =1 γ i − k p i − k,i γ i − λ i (cid:103) WC ( B i − k ) i ( ω ) , i = 1 , . . . , N. (3.14)As a consequence, the problem of finding (cid:102) W Ii ( · ) is reduced to finding (cid:103) WC ( B i − k ) i ( ω ) for all i, k = 1 , . . . , N .For notational reasons we first introduce the following N + 1 dimensional vectors, which willappear several times in this section: B G k , i = B , i if k < , B , i k − (cid:79) j =0 B j , i − if k = 1 , . . . , N, B N , i N − (cid:79) j =0 B j , i − if k = N, for i = 1 , . . . , N . Again, we use (cid:78) to denote the element-wise multiplication of vectors. Proposition 3.3
We have (cid:103) WC ( B i − k ) i ( ω ) = (cid:102) LC ( B i − k ) (cid:0) B G k , i (cid:1) k − (cid:89) j =0 (cid:101) R ∗ j,i − ( ω ) , (3.15)for i, k = 1 , . . . , N . Proof:
The key observation in the proof of Proposition 3.3 is that an arrival of an internallyrerouted customer always coincides with some service completion. For this reason, we considerthe system right after the service completion at, say, Q j ( j = 1 , . . . , N ). We compute thewaiting time LST of a customer routed to Q i after being served in Q j , conditional on thenumbers of customers of each type (now including gated customers) present at the arrivalepoch ( not including the arriving customer himself). We denote by n , . . . , n N , n G the eventthat the numbers of customers of all types are respectively n , . . . , n N , n G . Let n iG := n i if i (cid:54) = j , and n iG := n G if i = j . Note that the type G customers are located behind the gate in Q j , and that the customer routed to Q i only has to wait for these customers in case i = j .The waiting time of the tagged customer consists of:1. the service of all n j customers in front of the gate in Q j at the arrival epoch,10. the service of all n j +1 , . . . , n i − customers present in Q j +1 , . . . , Q i − at the arrival epoch,3. all of the descendants of the previously mentioned customers that will be served beforethe next visit to Q i ,4. if i (cid:54) = j , the service of all n iG customers present in Q i at the arrival epoch; if i = j , theservice of all n iG gated customers present in Q i at the arrival epoch,5. the switch-over times R j , . . . , R i − ,6. all customers arriving during these switch-over times that will be served before the nextvisit to Q i ,7. all of their descendants that will be served before the next visit to Q i .We denote the waiting time of an internal customer conditional on the event that he arrivesin Q i after being served in Q j , and conditional on the event that the numbers of customersof all types at the arrival epoch are respectively n , . . . , n N , n G , by WC ( B j ) (cid:48) i . Just like in theproof of Proposition 3.1, we can express WC ( B j ) (cid:48) i in terms of R , . . . , R N and S , . . . , S N : WC ( B j ) (cid:48) i = i − (cid:88) k = j n k (cid:88) l =1 S k l + R k + i − (cid:88) l = k +1 N l ( R k ) (cid:88) m =1 S l m + n iG (cid:88) l =1 B i,l . (3.16)Taking the LST of (3.16) leads to (3.15) after deconditioning. The derivation proceeds alongthe exact same lines as in the proof of Proposition 3.1, and is therefore omitted. (cid:3) External customers.
External customers arrive in Q i according to a Poisson process withintensity λ i . We distinguish between customers arriving during a switch-over time and cus-tomers arriving during a visit time. The waiting time of an external customer in Q i given that he arrives during R i − k is denoted by W ( R i − k ) i ( i, k = 1 , . . . , N ). Similarly, we use W ( V i − k ) i todenote an external customer arriving in Q i during V i − k . The waiting time LST of an arbitraryexternal customer can be expressed in terms of (cid:102) W ( R i − k ) i ( · ) and (cid:102) W ( V i − k ) i ( · ): (cid:102) W Ei ( ω ) = 1 E [ C ] N (cid:88) k =1 (cid:16) E [ V i − k ] (cid:102) W ( V i − k ) i ( ω ) + r i − k (cid:102) W ( R i − k ) i ( ω ) (cid:17) , i = 1 , . . . , N. (3.17)We first focus on the waiting time of customers arriving during a switch-over time. Considera tagged customer arriving in Q i during R i − k , i, k = 1 , . . . , N . Since the remaining part of theswitch-over time is part of the waiting time of the arriving customer, it will turn out that weneed the joint distribution of all customers present at the arrival epoch and the residual partof R i − k , denoted by R Ri − k . The PGF of the joint queue-length distribution at the arrival epochis given by (3.6). Equation (3.6) is based on the observation that the number of customers ineach queue at an arbitrary moment during R i − k is simply the sum of the number of customerspresent at the beginning of R i − k and the number of customers that have arrived during theelapsed (past) part of R i − k , denoted by R Pi − k . These random variables are independent.11ence, it is straightforward to adapt (3.6) to find the joint distribution of the queue lengths and residual part of R i − k , using the following result from elementary renewal theory: (cid:101) R P Rj ( ω P , ω R ) = (cid:101) R j ( ω P ) − (cid:101) R j ( ω R )( ω R − ω P ) r j , j = 1 , . . . , N, with (cid:101) R P Rj ( ω P , ω R ) denoting the LST of the joint distribution of past and residual switch-overtime R j . Hence, (cid:101) L ( R j ) ( z , ω ) = (cid:102) LB ( R j ) ( z ) (cid:101) R P Rj (Σ( z ) , ω ) , (3.18)where (cid:101) L ( R j ) ( z , ω ) denotes the PGF-LST of the joint distribution of the number of customersof each type at an arbitrary moment during R j and the residual part of R j . Obviously, thereare no gated customers present during a switch-over time.Consequently, and also using PASTA, we can find the waiting-time distribution by condi-tioning on the number of customers present at an arbitrary moment during R i − k and on theresidual switch-over time. Proposition 3.4
We have (cid:102) W ( R i − k ) i ( ω ) = (cid:101) R P Ri − k (cid:16) k − (cid:88) j =1 λ i − j (cid:0) − (cid:101) B ∗ j − ,i − ( ω ) (cid:1) + λ i (cid:0) − (cid:101) B i ( ω ) (cid:1) , ω + k − (cid:88) j =1 λ i − j (cid:0) − (cid:101) B ∗ j − ,i − ( ω ) (cid:1)(cid:17) × (cid:102) LB ( R i − k ) (cid:0) B G k − , i (cid:1) k − (cid:89) j =0 (cid:101) R ∗ j,i − ( ω ) , i, k = 1 , . . . , N, (3.19) Proof:
We consider an arbitrary customer arriving in Q i during R j . Similar to the proofs ofthe preceding propositions in this section, we condition on the number of customers presentin all queues at the arrival epoch, denoted by n , . . . , n N . As mentioned before, no gatedcustomers are present during a switch-over time. However, we also condition on the residuallength of R j , denoted by t R . The waiting time of the tagged customer consists of:1. the service of all n j +1 , . . . , n i − customers present at the arrival epoch in Q j +1 , . . . , Q i − ,2. the service of all their descendants that will be served before the start of the next visitto Q i ,3. the service of all n i customers present at the arrival epoch in Q i ,4. the residual switch-over time t R ,5. the switch-over times R j +1 , . . . , R i − ,6. the service of all customers arriving during t R , R j +1 , . . . , R i − that will be served beforethe start of the next visit to Q i ,7. the service of all descendants of these customers that will be served before the start ofthe next visit to Q i . 12f we denote the waiting time of a type i customer arriving during R j , conditional on n , . . . , n N and t R , by W ( R j ) (cid:48) i , we can summarise these items in the following formula: W ( R j ) (cid:48) i = i − (cid:88) k = j +1 n k (cid:88) l =1 S k l + R k + i − (cid:88) l = k +1 N l ( R k ) (cid:88) m =1 S l m + n i (cid:88) l =1 B i l + t R + i − (cid:88) l = j +1 N l ( t R ) (cid:88) m =1 S l m . (3.20)Taking the LST of (3.20) and using (3.18) leads to (3.19) after deconditioning. The derivationis not completely straightforward, but rather than providing it here, we refer to the proof ofProposition 3.5, which contains a similar derivation of a more complicated equation. (cid:3) Now we only need to determine (cid:102) W ( V i − k ) i ( · ). Focussing on a tagged customer arriving in Q i during the service of a customer in Q i − k , for i, k = 1 , . . . , N , we can find (cid:102) W ( V i − k ) i ( · ) byconditioning on the number of customers in each queue at the arrival epoch and the residualservice time. Similar to (cid:101) R P Rj ( · ), we define the LST of the joint distribution of past andresidual service time B j as (cid:101) B P Rj ( ω P , ω R ) = (cid:101) B j ( ω P ) − (cid:101) B j ( ω R )( ω R − ω P ) b j , j = 1 , . . . , N. (3.21)We can now use Equations (3.5) and (3.21) to find the PGF-LST of the joint distributionof the number of customers of each type present at an arbitrary moment during V j and theresidual service time of the customer that is being served at that moment: (cid:101) L ( V j ) ( z , ω ) = (cid:102) LB ( B j ) ( z ) (cid:101) B P Rj (Σ j ( z ) , ω ) . (3.22)Note that the customers arriving in Q j during the elapsed part of B j are gated customers. Proposition 3.5
We have (cid:102) W ( V i − k ) i ( ω ) = (cid:101) B P Ri − k (cid:16) k − (cid:88) j =1 λ i − j (cid:0) − (cid:101) B ∗ j − ,i − ( ω ) (cid:1) + λ i (cid:0) − (cid:101) B i ( ω ) (cid:1) , ω + k − (cid:88) j =1 λ i − j (cid:0) − (cid:101) B ∗ j − ,i − ( ω ) (cid:1)(cid:17) × (cid:102) LB ( B i − k ) (cid:0) B G k , i (cid:1) k − (cid:89) j =0 (cid:101) R ∗ j,i − ( ω ) × (cid:101) P ∗ k − ,i − ( ω ) (cid:101) B ∗ k − ,i − ( ω ) , (3.23)for i, k = 1 , . . . , N . Proof:
We denote by n , . . . , n N , n G the numbers of customers of all types present at thearrival epoch of the tagged customer. The residual part of the service time of the customerbeing served at this arrival epoch is denoted by t R . Let n iG := n i if i (cid:54) = j , and n iG := n G if i = j . The waiting time of a type i customer arriving during V j , conditional on n , . . . , n N , n G and the residual service time consists of the following components:1. the service of n j − Q j (We exclude the customerbeing served at the arrival epoch),2. the service of all n j +1 , . . . , n i − customers present in Q j +1 , . . . , Q i − ,13. all of the descendants of the previously mentioned customers that will be served beforethe next visit to Q i ,4. if i (cid:54) = j , the service of all n iG customers present in Q i at the arrival epoch; if i = j , theservice of all n iG gated customers present in Q i ,5. the switch-over times R j , . . . , R i − ,6. the residual service time t R ,7. all customers arriving during t R and R j , . . . , R i − that will be served before the nextvisit to Q i ,8. all of their descendants that will be served before the next visit to Q i ,9. the (possible) future service of the customer being served at the arrival epoch, due tothe fact that he may be routed to another queue that will be served before the nextvisit to Q i ,10. the service of all descendants of this rerouted customer (Note that if he will be reroutedand served again, he will count as his own descendant).More formally: W ( V j ) (cid:48) i = n j − (cid:88) l =1 S j,l + i − (cid:88) k = j +1 n k (cid:88) l =1 S k l + n iG (cid:88) l =1 B i l + i − (cid:88) k = j R k + i − (cid:88) l = k +1 N l ( R k ) (cid:88) m =1 S l m + t R + i − (cid:88) l = j +1 N l ( t R ) (cid:88) m =1 S l m + (cid:40) S l for l = j + 1 , . . . , i −
1, w.p. p j,l , − (cid:80) i − l = j +1 p j,l , . (3.24)14e now show that Equation (3.23) follows from taking the LST: E [e − ωW ( Vj ) i | n , . . . , n N , n iG ]= E n j − (cid:89) l =1 e − ωS jl i − (cid:89) m = j +1 n m (cid:89) l =1 e − ωS ml E (cid:34) n iG (cid:89) l =1 e − ωB il (cid:35) E i − (cid:89) m = j e − ω (cid:16) R m + (cid:80) i − l = m +1 (cid:80) Nl ( Rm ) q =1 S lq (cid:17) × e − ωt R E i − (cid:89) l = j +1 N l ( t R ) (cid:89) m =1 e − ωS lm i − (cid:88) l = j +1 p j,l E (cid:2) e − ωS l (cid:3) + 1 − i − (cid:88) l = j +1 p j,l = E (cid:2) e − ωS j (cid:3) n j − i − (cid:89) m = j +1 E (cid:2) e − ωS m (cid:3) n m E (cid:2) e − ωB i (cid:3) n iG i − (cid:89) m = j (cid:101) R m (cid:16) ω + i − (cid:88) l = m +1 (1 − E [e − ωS l ]) (cid:17) × e − ωt R i − (cid:89) l = j +1 ∞ (cid:88) m =0 E [e − ωS l ] m P [ N l ( t R ) = m ] − i − (cid:88) l = j +1 p j,l (cid:16) − E (cid:2) e − ωS l (cid:3) (cid:17) = (cid:101) B ∗ k − ,i − ( ω ) n i − k − k − (cid:89) l =1 (cid:101) B ∗ l − ,i − ( ω ) n i − l (cid:101) B i ( ω ) n iG k (cid:89) l =1 (cid:101) R ∗ l − ,i − ( ω ) × exp − (cid:16) ω + i − (cid:88) l = j +1 (1 − E [e − ωS l ]) (cid:17) t R (cid:101) P ∗ k − ,i − ( ω )= (cid:101) B ∗ k − ,i − ( ω ) n i − k k − (cid:89) l =1 (cid:101) B ∗ l − ,i − ( ω ) n i − l (cid:101) B i ( ω ) n iG k (cid:89) l =1 (cid:101) R ∗ l − ,i − ( ω ) × exp (cid:34) − (cid:16) ω + k − (cid:88) l =1 (1 − (cid:101) B ∗ l − ,i − ( ω )) (cid:17) t R (cid:35) P k − ,i − ( ω ) (cid:101) B ∗ k − ,i − ( ω ) , where k = i − j (or k = N + i − j if j ≥ i ). Deconditioning of this expression leads to (3.23). (cid:3) Arbitrary customers.
Finally, we present the main result of this section: the LST of thewaiting-time distribution of an arbitrary customer in Q i . Theorem 3.6
The LST of the waiting-time distribution of an arbitrary customer in Q i , ifthis queue receives gated service, is given by: (cid:102) W i ( ω ) = γ i − λ i γ i (cid:102) W Ii ( ω ) + λ i γ i (cid:102) W Ei ( ω ) , i = 1 , . . . , N, (3.25)where (cid:102) W Ii ( ω ) and (cid:102) W Ei ( ω ) are given by (3.14) and (3.17), respectively. Proof:
The result follows immediately after conditioning on the event that an arbitrarycustomer is an internal or external customer. (cid:3) Exhaustive service
In this section we study systems with mixtures of gated and exhaustive service, that is,some queues are served exhaustively whereas other queues receive gated service. We restrictourselves to presenting the results, but for reasons of compactness we omit all proofs as theycan be produced similar to the proofs in the previous section.Throughout we use the index e ∈ { , . . . , N } to refer to an arbitrary queue with exhaustiveservice, which means that customers are being served until the queue is empty. This meansthat, in contrast to gated service, customers arriving in Q e during V e will be served duringthat same visit period. This is true, even if the customer has just received service in Q e andwas routed back to Q e again. To deal with this issue, we define an extended service time B exh e which is the total amount of service that a customer receives during a visit period V e before being routed to another queue (or leaving the system), cf. [29]. As stated in [29], B exh e is the geometric sum, with parameter p e,e , of independent random variables with the samedistribution as B e . The LST of B exh e is given by (cid:101) B exh e ( ω ) = (1 − p e,e ) (cid:101) B e ( ω )1 − p e,e (cid:101) B e ( ω ) . We denote a busy period of type e customers by BP e . The PGF-LST of the joint distributionof a busy period and the number of customers served during this busy period satisfies thefollowing equation: (cid:102) BP e ( z, ω ) = z (cid:101) B exh e (cid:0) ω + λ e (1 − (cid:102) BP e ( z, ω )) (cid:1) . Visit beginnings and completions.
The laws of motion (3.1a)-(3.1e) have to be adaptedif a queue receives exhaustive service. First we need to redefine Σ i ( z ) and P i ( z ) if Q i is servedexhaustively, and introduce P exh i ( z ):Σ e ( z ) = (cid:88) j (cid:54) = e λ j (1 − z j ) ,P e ( z ) = p e, + N (cid:88) j =1 p e,j z j ,P exh e ( z ) = p e, − p e,e + (cid:88) j (cid:54) = e p e,j − p e,e z j , for all e ∈ { , . . . , N } corresponding to queues with exhaustive service. The laws of motionnow change accordingly: (cid:102) LC ( V e ) ( z ) = (cid:102) LB ( V e ) (cid:16) z , . . . , z e − , (cid:102) BP e (cid:0) P exh e ( z ) , Σ e ( z ) (cid:1) , z e +1 , . . . , z N , (cid:17) , (cid:102) LB ( R e ) ( z ) = (cid:102) LC ( V e ) ( z ) , for any exhaustively served Q e . 16 ervice beginnings and completions. Eisenberg’s relation (3.2) remains valid for queueswith exhaustive service. Note that P e ( z ) should not be replaced by P exh e ( z ) for exhaustivequeues in (3.2)! Relation (3.3) should be slightly changed for queues with exhaustive service,since customers are not placed behind a gate: (cid:102) LC ( B e ) ( z ) = (cid:102) LB ( B e ) ( z ) (cid:101) B e (cid:0) Σ( z ) (cid:1) /z e . Arbitrary moments.
Equation (3.4) for the PGF of the joint queue-length distributionat arbitrary moments remains valid if some of the queues have exhaustive service. However, (cid:101) L ( V j ) ( z ) should be adapted for queues with exhaustive service by replacing gated customerswith “ordinary” type e customers: (cid:101) L ( V e ) ( z ) = (cid:102) LB ( B e ) ( z ) 1 − (cid:101) B e (cid:0) Σ( z ) (cid:1) b e Σ( z ) . The fact that customers arriving in an exhaustively served queue, say Q i − k , during V i − k areserved before the end of this visit period, requires changes in the definition of (cid:101) B ∗ k,i ( ω ). (cid:101) B ∗ k,i ( ω ) = (cid:102) BP i − k (cid:16) (cid:101) P ∗ k,i ( ω ) , ω + k − (cid:88) j =0 λ i − j (cid:0) − (cid:101) B ∗ j,i ( ω ) (cid:1)(cid:17) , k = 0 , , . . . , N ; i = 1 , . . . , N, (4.1)where (cid:101) P ∗ k,i ( ω ) = 1 − k − (cid:88) j =0 p i − k,i − j − p i − k,i − k (cid:0) − (cid:101) B ∗ j,i ( ω ) (cid:1) , k = 0 , , . . . , N ; i = 1 , . . . , N. (4.2)Given this modified definition of (cid:101) B ∗ k,i ( ω ), the function (cid:101) R ∗ k,i ( ω ) remains unchanged. The ex-pression for the LST of the cycle time C i , given by (3.12), also remains valid for systemscontaining exhaustively served queues. Internal customers.
The waiting time LST of internal customers (3.14) is determined byconditioning on the event that an arrival in Q i follows a service completion in some Q i − k . Asstated before, for queues with exhaustive service we need to take into account that customersthat are routed back to the same queue will be served during the same visit period. For anarbitrary exhaustively served queue Q e , this results in (cid:102) W Ie ( ω ) = N − (cid:88) k =0 γ e − k p e − k,e γ e − λ e (cid:103) WC ( B e − k ) e ( ω ) . (4.3)Compared to (3.14), the summation starts at k = 0 and runs up to k = N −
1. We nowintroduce B (cid:48) , i = (cid:0) , . . . , , (cid:101) B i ( ω ) , , . . . , (cid:1) , i = 1 , . . . , N, (cid:101) B i ( ω ) at the position corresponding to customers in Q i . If Q i has exhaustive service,there is a subtle difference with B , i which has (cid:102) BP i (1 , ω ) at position i . We can now determine (cid:103) WC ( B e − k ) e ( ω ) for any Q e that receives exhaustive service: (cid:103) WC ( B e − k ) e ( ω ) = (cid:102) LC ( B e − k ) (cid:0) B (cid:48) , e k − (cid:79) j =0 B j , e − (cid:1) k − (cid:89) j =0 (cid:101) R ∗ j,e − ( ω ) , k = 1 , . . . , N − , (cid:103) WC ( B e ) e ( ω ) = (cid:102) LC ( B e ) (cid:0) B (cid:48) , e (cid:1) . For each Q i that receives gated service, we can still use (3.14) with the modified definition of (cid:101) B ∗ k,i ( ω ) for each Q i − k which receives exhaustive service. External customers.
The waiting time LST of external customers (3.17) is determinedby conditioning on the event that an arrival in Q i takes place during V i − , . . . , V i − N or dur-ing R i − , . . . , R i − N . Before discussing the waiting times of external customers arriving inan exhaustively served queue, it is important to realise that allowing some queues to haveexhaustive service will now also require some changes to waiting times of customers arrivingin a queue with gated service. This means that (3.23) should now become (cid:102) W ( V i − k ) i ( ω ) = (cid:101) B P Ri − k (cid:16) k − (cid:88) j =1 λ i − j (cid:0) − (cid:101) B ∗ j − ,i − ( ω ) (cid:1) + λ i (cid:0) − (cid:101) B i ( ω ) (cid:1) + λ i − k (cid:0) − (cid:101) B ∗ k − ,i − ( ω ) (cid:1) ,ω + k − (cid:88) j =1 λ i − j (cid:0) − (cid:101) B ∗ j − ,i − ( ω ) (cid:1) + λ i − k (cid:0) − (cid:101) B ∗ k − ,i − ( ω ) (cid:1)(cid:17) × (cid:102) LB ( B i − k ) (cid:0) B , i k − (cid:79) j =0 B j , i − (cid:1) k − (cid:89) j =0 (cid:101) R ∗ j,i − ( ω ) × − (cid:80) k − j =0 p i − k,i − j − (cid:0) − (cid:101) B ∗ j,i − ( ω ) (cid:1)(cid:101) B ∗ k − ,i − ( ω ) , (4.4)if Q i − k receives exhaustive service (and Q i receives gated service). Compared to (3.23) wecan see that there are two additional terms λ i − k (cid:0) − (cid:101) B ∗ k − ,i − ( ω ) (cid:1) which take into accountthat customers arriving in Q i − k during the elapsed and during the residual part of the presentservice time B i − k will be served during the present visit period. Furthermore, we can see that (cid:101) P ∗ k − ,i − ( ω ) has been replaced by 1 − (cid:80) k − j =0 p i − k,i − j − (cid:0) − (cid:101) B ∗ j,i − ( ω ) (cid:1) , which is required becausethe customer being served should be allowed to return to Q i − k upon his service completion.If Q e receives exhaustive service we have to make some additional changes. We have (cid:102) W Ee ( ω ) = 1 E [ C ] N (cid:88) k =1 (cid:16) E [ V e − k +1 ] (cid:102) W ( V e − k +1 ) e ( ω ) + r e − k (cid:102) W ( R e − k ) e ( ω ) (cid:17) , (4.5)where we have chosen to denote the waiting time LST of customers arriving in Q e during V e as (cid:102) W ( V e ) e ( ω ) rather than (cid:102) W ( V e − N ) e ( ω ) to illustrate the fact that they will be served during thesame visit period. The expression for (cid:102) W ( R e − k ) e ( ω ), given by (3.19), should be slightly modifiedif Q e receives exhaustive service. However, since the only required modification is that B , i should be replaced by B (cid:48) , i , we refrain from giving the complete expression.18f k >
0, the expression for (cid:102) W ( V e − k ) e ( ω ) remains almost the same as (3.23) if Q e − k receivesgated service, or (4.4) if Q e − k receives exhaustive service. The only change is, once again,that B , i should be replaced by B (cid:48) , i . The case k = 0 results in a much simpler expression,since we only have to wait for the service times of the customers that were present at thebeginning of the present service (excluding the customer in service) plus the service times ofthe customers that have arrived in Q e during the elapsed part of the present service, plus theresidual service time: (cid:102) W ( V e ) e ( ω ) = (cid:101) B P Re (cid:16) λ e (cid:0) − (cid:101) B e ( ω ) (cid:1) , ω (cid:17) (cid:102) LB ( B e ) (cid:0) B (cid:48) , e (cid:1)(cid:101) B e ( ω ) . Arbitrary customers.
The LST of the waiting-time distribution of an arbitrary customerin an exhaustively served queue immediately follows after conditioning on the event that anarbitrary customer is either an internal or an external customer, similar to the derivationof (3.25). The result is presented in the theorem below.
Theorem 4.1
The LST of the waiting-time distribution of an arbitrary customer in Q i , ifthis queue receives exhaustive service, is given by: (cid:102) W i ( ω ) = γ i − λ i γ i (cid:102) W Ii ( ω ) + λ i γ i (cid:102) W Ei ( ω ) , i = 1 , . . . , N, (4.6)where (cid:102) W Ii ( ω ) and (cid:102) W Ei ( ω ) are defined in (4.3) and (4.5). In this section we give some numerical examples that indicate the versatility of the modelthat we have discussed. To this end, we use some examples that can be found in the existingliterature, and show how our model can be used to describe the various systems and find therelevant performance measures. Hence, most of the results presented in this section are notnovel, but the way of deriving them is new.
Example 1: tandem queues with parallel queues in the first stage.
We first use anexample that was introduced by Katayama [19], who studies a network consisting of threequeues. Customers arrive at Q and Q , and are routed to Q after being served (see Figure1). This model, which is referred to as a tandem queueing model with parallel queues inthe first stage, is a special case of the model discussed in the present paper. We simplyput p , = p , = p , = 1 and all other p i,j are zero. We use the same values as in [19]: λ = λ /
10, service times are deterministic with b = b = 1, and b = 5. The server servesthe queues exhaustively, in cyclic order: 1, 2, 3, 1, . . . . The only difference with the modeldiscussed in [19] is that we introduce (deterministic) switch-over times R = R = 2. Weassume that no time is required to switch between the two queues in the first stage, so r = 0.In Figure 2 we show the means and standard deviations of the waiting times of customersat the three queues. These plots reveal that in the heavy-traffic regime, as ρ ↑
1, the mean waiting times of customers in Q are close to those in Q , but the standard deviations of19 Q Q λ λ Server
Figure 1: Tandem queues with parallel queues in the first stage, as discussed in Example 1.the waiting times in Q are closer to those in Q . Further inspection of the exact results,obtained by differentiating the LSTs, confirms that in both cases the limits are very close,but not exactly the same.It is also interesting to study the light-traffic behaviour of the system, i.e., as ρ ↓
0. Fromthe plots in Figure 2 we can see that, as ρ ↓
0, the mean waiting times are all equal, but the standard deviation of the waiting times in Q and Q is different than in Q . From the LSTsof the waiting-time distributions we can obtain the exact expressions when ρ ↓
0, by takingthe Taylor expansion in ρ at ρ = 0 and subsequently ignoring all O ( ρ ) terms. This, combinedwith the fact that R = 0 and all of the routing probabilities are either 0 or 1, considerablysimplifies all expressions from the previous section: (cid:102) W ( ω ) = (cid:102) W E ( ω ) → r r (cid:102) W ( R )1 ( ω ) + r r (cid:102) W ( R )1 , (cid:102) W ( ω ) = (cid:102) W E ( ω ) → r r (cid:102) W ( R )2 ( ω ) + r r (cid:102) W ( R )2 , (cid:102) W ( ω ) = (cid:102) W I ( ω ) → λ λ + λ (cid:103) WC ( B )3 ( ω ) + λ λ + λ (cid:103) WC ( B )3 ( ω ) . Since we are considering the case ρ ↓
0, these expressions can be simplified even further toclosed-form expressions, because ignoring all O ( ρ ) terms is equivalent to regarding the systemas being empty all the time: (cid:102) W ( ω ) → r r (cid:101) R P R (0 , ω ) (cid:101) R ( ω ) + r r (cid:101) R P R (0 , ω ) , (cid:102) W ( ω ) → r r (cid:101) R P R (0 , ω ) (cid:101) R ( ω ) + r r (cid:101) R P R (0 , ω ) , (cid:102) W ( ω ) → λ λ + λ (cid:101) R ( ω ) (cid:101) R ( ω ) + λ λ + λ (cid:101) R ( ω ) . These expressions reveal the true behaviour of the system in light traffic. The waiting timesin Q and Q are simply the total residual switch-over time, with mean r (2) / r = 2 andsecond moment r (3) / r = 16 /
3. For queue Q the situation is different, because this queueonly contains internally rerouted customers. Customers being rerouted from Q have to waitfor the switch-over times R + R , whereas customers arriving from Q have to wait only for20 . Since R = 0, the waiting time only consists of R = 2 in both cases. Substituting allparameter values results in the following LT limits of the waiting-time LSTs: (cid:102) W ( ω ) → − e − ω ω , (cid:102) W ( ω ) → − e − ω ω , (cid:102) W ( ω ) → e − ω ( ρ ↓ . Differentiating the LSTs gives the following results as ρ ↓ E [ W ] → , E [ W ] → , E [ W ] → , sd[ W ] → (cid:112) / , sd[ W ] → (cid:112) / , sd[ W ] → . E [ W ] E [ W ] E [ W ] ρ sd [ W ] sd [ W ] sd [ W ] ρ Figure 2: Means and standard deviations of the waiting times in the first numerical example.
Example 2: a two-stage queueing model with customer feedback.
This secondexample is introduced by Tak´acs [30], and extended by Ali and Neuts [1]. The queueingsystem under consideration consists of a waiting room, in which customers arrive accordingto a Poisson process with intensity λ , and a service room. The customers are all transferredsimultaneously to the service room where they receive service in order of arrival. However,at the moment of the transfer to this service room M additional “overhead customers” areadded to the front of this queue. (In [30] M is a constant, in [1] it is a random variable.) Uponservice completion, each customer leaves the system with probability q , and returns to thewaiting room with probability 1 − q . Overhead customers leave the system with probabilityone after being served. As soon as the last customer in the service room finishes service (andeither leaves the system, or returns to the waiting room) all customers present in the waitingroom are transferred to the service room, where they will receive service after a new batch ofoverhead customers has been served, and so on. A schematic representation of this model isdepicted in Figure 3.We use the same input parameters as Tak´acs [30]: q = 2 / λ/µ = 1 /
6, where 1 /µ is themean service time in the service room. This service time is exponentially distributed. Thenumber of overhead customers that are added to the front of the queue is a constant withvalue M . We can model this system in terms of our network with a single, shared serverby defining arrival intensities λ = λ and λ = 0. The service times in stations 1 and 2 arerespectively 0 and exponentially distributed with mean b = 1 /µ . The routing probabilitiesare p , = 1 and p , = 1 /
3, the other p i,j are zero. The service times of the overhead21 Waiting room Service room1 − q q Server M Figure 3: The two-stage queueing model with customer feedback, as discussed in Example 2.customers are also exponentially distributed with parameter µ . Hence, we can model theaddition of M overhead customers as a switch-over time which is Erlang- M distributed withparameter µ . The switch-over time between Q and Q is zero. Note that, since b = 0,there is no difference between gated and exhaustive service. By differentiation of the waitingtime LSTs (3.25), we can obtain explicit expressions for all moments of the waiting-timedistributions for this example. The first three moments of the waiting times are given below. E [ W ] = 1 + M µ , E [ W ] = ( M + 1)(11 M + 25)27 µ , E [ W ] = ( M + 1)( M (43 M + 223) + 310)108 µ , E [ W ] = 1 + 7 M µ , E [ W ] = ( M + 1)(37 M + 11)27 µ , E [ W ] = ( M + 1)( M + 2)(175 M + 81)108 µ . The results are slightly different from those presented in [30], because Tak´acs also considersthe overhead customers in the computations of the waiting times and allows them to returnto the waiting room after their service is completed. Modelling this situation would requireone minor adaptation in the laws of motion (adding the overhead customers at the beginningof V ) and another adaptation in the waiting time LST (conditioning on the event that a newcustomer is an overhead customer). These changes are not too difficult but beyond the scopeof this paper. In this section, we not only elaborate on the developed method and its applicability, but wealso discuss possible ways of extending the present study.
Method.
As mentioned in the introduction, the main complicating factor of the modelunder consideration is caused by the rerouting of internal customers. This implies that the total arrival process at each queue is not Poisson, and not even renewal. Traditional methodsto determine waiting-time distributions in each queue are based on the distributional form ofLittle’s Law, which relies on the assumption of Poisson arrivals. Contrary to the distributionalform of Little’s Law, we explicitly make use of the branching structure to find waiting-timedistributions. The main idea is that upon the arrival of a tagged customer Y at time t at Q i we compute a priori the total future service times at each of the queues, for all the othercustomers present in the system at time t that will be served before customer Y enters service22t Q i (see (3.7)). Additionally, we add the total future service requirements of all externalarrivals (and their descendants) that will be served before customer Y enters service (see(3.9)). The advantage of this method is that a system no longer needs to satisfy all of theprerequisites required to apply the distributional form of Little’s Law (see [21]). Applicability.
The novel approach of this paper to find the LST of the waiting-time dis-tribution can also be applied to other types of models with a single server serving multiplequeues. Obviously, one can apply it to standard polling models (without customer routing)by simply taking p i, = 1 and p i,j = 0 for j >
0. However, the developed methodology carriesalmost directly over to tandem queues [23, 34], multi-stage queueing models with parallelqueues [19], feedback vacation queues [9, 33], symmetric feedback polling systems [31, 33],systems with a waiting room [1, 30], closed networks [2],
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Further research.
Since the model can be described as a multi-type branching process, ex-plicit closed-form expressions can be obtained for the waiting-time distributions under heavy-traffic (HT) assumptions. Such expressions are appealing because they give fundamentalinsight in how the system performance depends on the system parameters, and in particularon the routing probabilities p i,j . HT asymptotics can be obtained by combining insights frommulti-type branching processes [35], fluid analyses [24, 25], and the heavy-traffic averagingprinciple by Coffman et al. [12, 13]. The HT analysis is relevant because in practice theproper operation of the system is particularly important when the system is heavily loaded.The HT asymptotics form an excellent basis for the development of approximations for thewaiting-time distributions for arbitrary loads. For the mean waiting times, preliminary resultsare obtained in [6].From a practical perspective, motivated by applications in production systems [5], an impor-tant extension of the model under consideration is a model where customers visit a prede-termined, class-specific sequence of queues in a fixed order. In our model one would have todefine multiple customer classes, each having their own fixed visit order through the system.The method presented in this paper forms a good basis for this extension. Acknowledgements
The authors are grateful to Ivo Adan and Onno Boxma for providing valuable comments onearlier drafts of the present paper.
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