Mixed Gated/Exhaustive Service in a Polling Model with Priorities
MMixed Gated/Exhaustive Service in a Polling Model withPriorities ∗ M.A.A. Boon † [email protected] I.J.B.F. Adan † [email protected], 2009 Abstract
In this paper we consider a single-server polling system with switch-over times. Weintroduce a new service discipline, mixed gated/exhaustive service, that can be used forqueues with two types of customers: high and low priority customers. At the beginningof a visit of the server to such a queue, a gate is set behind all customers. High prioritycustomers receive priority in the sense that they are always served before any low prioritycustomers. But high priority customers have a second advantage over low priority cus-tomers. Low priority customers are served according to the gated service discipline, i.e.only customers standing in front of the gate are served during this visit. In contrast, highpriority customers arriving during the visit period of the queue are allowed to pass thegate and all low priority customers before the gate.We study the cycle time distribution, the waiting time distributions for each customertype, the joint queue length distribution of all priority classes at all queues at pollingepochs, and the steady-state marginal queue length distributions for each customer type.Through numerical examples we illustrate that the mixed gated/exhaustive service disci-pline can significantly decrease waiting times of high priority jobs. In many cases thereis a minimal negative impact on the waiting times of low priority customers but, remark-ably, it turns out that in polling systems with larger switch-over times there can be evena positive impact on the waiting times of low priority customers.
Keywords:
Polling, priority levels, queue lengths, waiting times, mixed gated/exhaustive
There are three ways in which one can introduce prioritisation into a polling model. The firsttype of priority is by changing the server routing such that certain queues are visited morefrequently than other queues [6, 19]. This type of prioritisation is quite common in wirelessnetwork protocols. A second type of prioritisation is through differentiation of the number ofcustomers that are served during each visit to a queue. This type of prioritisation is inflicted ∗ The research was done in the framework of the BSIK/BRICKS project, and of the European Network ofExcellence Euro-FGI. † 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 hrough the usage of different service disciplines. For example, one can serve all customers ina queue before switching to the next queue (exhaustive service), or one can limit the amountof customers that are served to, e.g., only those customers present at the arrival of the serverat the queue (gated service). Typically, this will have a negative impact on the waiting timesof the customers in queues that are not served exhaustively. The third way of introducingpriorities is by changing the order in which customers are served within a queue, which is apopular technique to improve performance of production systems, cf. [2, 22]. The presentpaper introduces a new service discipline, referred to as mixed gated/exhaustive service, thatcombines the last two types of prioritisation.In the polling model considered in the present paper a single server visits N queues in afixed, cyclic order. Some, or even all, of the queues contain two types of customers: highand low priority customers. For these queues we introduce a new service discipline, calledmixed gated/exhaustive service based on the priority level of the customer. A polling systemwith high and low priority customers in a queue with purely gated or exhaustive service hasbeen studied in [1, 2]. The mixed gated/exhaustive service discipline can be considered asa mixture of these two service disciplines where low priority customers receive gated serviceand high priority customers receive exhaustive service. A more detailed description is givenin Section 2. Since the number of customers served during one visit in a queue with gatedservice is different from the number served during a visit with exhaustive service, the mixedgated/exhaustive service discipline introduced in the present paper combines the second andthe third type of prioritisation. A variation of the model under consideration, namely a pollingsystem where low priority customers are served only if there are no high priority customerspresent in any of the queues , has been studied in [12].Polling models have been studied for many years and because of their practical relevancemany papers on polling systems have been written in a mixture of application areas. Thesurvey of Takagi [21] on polling systems and their applications from 1991 is still very valuable,although the last couple of years interest in polling models has revived, partly triggered bymany new applications. The motivation for the present paper is to present a service disciplinethat combines the benefits of the gated and exhaustive service disciplines for priority pollingmodels. The specific application that attracted our attention is in the field of logistics.Consider a make-to-order production system with a single production capacity for multipleproducts. In many firms encountering this situation, the products are produced according toa fixed production sequence. The production capacity, where the production orders queueup, can be represented as a polling model by identifying each product with a queue and thedemand process of a product with the arrival process at the corresponding queue. For a moredetailed description of fixed-sequence strategies in the context of make-to-stock productionsituations, see [23]. In the context of this production setting, the situation with two ormore priority levels - as studied in detail in the present paper - is oftentimes encountered inpractice, where production departments have to supply both internal and external customers,the latter of which is commonly given a preferential treatment. A different application stemsfrom production scheduling in flexible manufacturing systems where part types are oftengrouped with other types sharing (almost) similar characteristics, such that no change ofmachine configuration, i.e. setup time, is required when switching between these part types(see, e.g., [17]). Since no setup time is required to switch between these types, it can be seenas customers of different types being served in the same queue. The introduction of prioritiescan be useful to efficiently differentiate between different parts grouped within one queue.2hese two applications make the practical relevance of the inclusion of multiple priority levelsin the studied polling model evident. Finally, we should keep in the back of our mind that theresults of the present paper are certainly not limited to these production settings, but may beused in many other fields where polling models arise, such as communication, transportationand health care (e.g., surgery procedures where an urgency parameter is assigned to eachpatient).The present paper is structured as follows: first we discuss the model in more detail andwe determine the generating functions (GFs) of the joint queue length distribution of allcustomers at visit beginnings and completions of each queue. In Section 4 we determinethe Laplace-Stieltjes Transforms (LSTs) of the distributions of the cycle time, visit timesand intervisit times. These distributions are used to determine the marginal queue lengthdistributions and waiting time distributions of high and low priority customers in all queues.The LST of the waiting time distribution is used to compute the mean waiting time of eachcustomer type. A pseudo-conservation law for these mean waiting times is presented inSection 7. Furthermore, we introduce some numerical examples to illustrate typical featuresof a polling model with mixed gated/exhaustive service. Finally we discuss possible extensionsand future research on the topic. The model considered in the present paper is a polling model which consists of N queues,labelled Q , . . . , Q N . Throughout the whole paper all indices are modulo N , so Q N +1 standsfor Q . The queues are visited by one server in a fixed, cyclic order: 1 , , . . . , N, , , . . . .The switch-over time of the server from Q i to Q i +1 is denoted by S i with LST σ i ( · ). Weassume that all switch-over times are independent and at least one switch-over time is strictlygreater than zero. Each queue contains two customer types: high and low priority customers,although the analysis allows any number (greater than zero) of customer types per queue.High priority customers in Q i are called type iH customers and low priority customers in Q i are called iL customers, i = 1 , . . . , N . Type iH customers arrive at Q i according to aPoisson process with intensity λ iH , and type iL customers arrive at Q i according to a Poissonprocess with intensity λ iL . The service times of type iH and iL customers are denoted by B iH and B iL , with LSTs β iH ( · ) and β iL ( · ). All service times are assumed to be independent.We introduce the notation ρ iH = λ iH E ( B iH ) and similarly ρ iL = λ iL E ( B iL ). The totaloccupation rate of the system is ρ = (cid:80) Ni =1 ρ i , where ρ i = ρ iH + ρ iL is the fraction of timethat the server visits Q i . Service of the customers is gated for low priority customers andexhaustive for high priority customers. In more detail: each queue actually contains twolines of waiting customers: one for the low priority customers and one for the high prioritycustomers. At the beginning of a visit to Q i , a gate is set behind the low priority customersto mark them eligible for service. High priority customers are always served exhaustivelyuntil no high priority customer is present. When no high priority customers are presentin the queue, the low priority customers standing in front of the gate are served in orderof arrival, but whenever a high priority customer enters the queue, he is served before anywaiting low priority customers. Service is non-preemptive though, implying that service ofa type iL customer is not interrupted by an arriving type iH customer. The visit to Q i ends when all type iL customers present at the beginning of this visit are served and no high3riority customers are present in the queue. Notice that if the arrival intensity λ iH equals 0,then Q i is served completely according to the gated service discipline. Similarly we can set λ iL = 0 to obtain a purely exhaustively served queue. Both the gated and the exhaustiveservice discipline fall into the category of branching-type service disciplines. These are servicedisciplines that satisfy the following property, introduced by Resing [16] and Fuhrmann [10]. Property 2.1
If the server arrives at Q i to find k i customers there, then during the courseof the server’s visit, each of these k i customers will effectively be replaced in an i.i.d. mannerby a random population having probability generating function h i ( z , . . . , z N ), which can beany N -dimensional probability generating function.If Q i receives gated service, we have h i ( z , . . . , z N ) = β i (cid:16)(cid:80) Nj =1 λ j (1 − z j ) (cid:17) , where β i ( · ) de-notes the service time LST of an arbitrary customer in Q i , and λ i denotes his arrival rate.For exhaustive service h i ( z , . . . , z N ) = π i (cid:16)(cid:80) j (cid:54) = i λ j (1 − z j ) (cid:17) , where π i ( · ) is the LST of a busyperiod distribution in an M/G/ i customers, so it is the root in (0 , π i ( ω ) = β i ( ω + λ i (1 − π i ( ω ))), ω ≥ Q i receives mixed gated/exhaustive service, because the randompopulation that replaces each of these customers depends on the priority level. In the nextsection we circumvent this problem by splitting each queue into two virtual queues, eachof which has a branching-type service discipline. This equivalent polling system satisfiesProperty 2.1, so we can still use the methodology described in [16] to find, e.g., the joint queuelength distribution at visit beginnings and completions. All other probability distributionsthat are derived in the present paper can be expressed in terms of (one of) these joint queuelength distributions. In the present section we analyse a polling system with all queues having two priority levelsand receiving mixed gated/exhaustive service, but in fact each queue would be allowed to haveany branching-type service discipline. Denote the GF of the joint queue length distribution oftype 1 H, L, . . . , N H, N L customers at the beginning and the completion of a visit to Q i byrespectively V b i ( z H , z L , . . . , z NH , z NL ) and V c i ( z H , z L , . . . , z NH , z NL ). As discussed in theprevious section, the polling model under consideration does not satisfy Property 2.1, whichoften means that an exact analysis is difficult or even impossible. For this reason we introducea different polling system that does satisfy Property 2.1 and has the same joint queue lengthdistribution at visit beginnings and endings. The equivalent system contains 2 N queues,denoted by Q H ∗ , Q L ∗ , . . . , Q NH ∗ , Q NL ∗ . The switch-over times S i , i = 1 , . . . , N , are incurredwhen the server switches from Q iL ∗ to Q ( i +1) H ∗ ; there are no switch-over times between Q iH ∗ and Q iL ∗ . Customers in this system are so-called “smart customers”, introduced in [4],meaning that the arrival rate of each customer type depends on the location of the server.Type iH ∗ customers arrive in Q iH ∗ according to arrival rate λ iH unless the server is serving Q iL ∗ . When the server is serving Q iL ∗ , the arrival rate of type iH ∗ customers is 0. Thereason for this is that we incorporate the service times of all type iH customers that wouldhave arrived during the service of a type iL customer, in the original polling model, into theservice time of a type iL ∗ customer. In our alternative system, type iL ∗ customers arrive with4ntensity λ iL and have service requirement B ∗ iL with LST β ∗ iL ( · ). There is a simple relationbetween B iL and B ∗ iL , expressed in terms of the LST: β ∗ iL ( ω ) = β iL ( ω + λ iH (1 − π iH ( ω ))) . (3.1) B ∗ iL is often called completion time in the literature, cf. [20], with mean E ( B ∗ iL ) = E ( B iL )1 − ρ iH . Ser-vice is exhaustive for Q H ∗ , Q H ∗ , . . . , Q NH ∗ and synchronised gated for Q L ∗ , Q L ∗ , . . . , Q NL ∗ ,the gate of Q iL ∗ being set at the visit beginning of Q iH ∗ . The synchronised gated servicediscipline is introduced in [15] and does not strictly satisfy Property 2.1. However, it does satisfy a slightly modified version of Property 2.1 that still allows for straightforward analysis;see [3] for more details. During a visit to Q iL ∗ only those type iL ∗ customers are served thatwere present at the previous visit beginning to Q iH ∗ . The joint queue length distribution ata visit beginning of Q iH ∗ in this system is the same as the joint queue length distributionat a visit beginning of Q i in the original polling system. Similarly, the joint queue lengthdistribution at a visit completion of Q iL ∗ is the same as the joint queue length distributionat a visit completion of Q i in the original polling system. In terms of the GFs: V b i ( z ) = V b iH ∗ ( z ) ,V c i ( z ) = V c iL ∗ ( z ) , where z is a shorthand notation for the vector ( z H , z L , . . . , z NH , z NL ). The GFs of the jointqueue length distributions at a visit beginning and completion of Q iH ∗ are related in thefollowing manner: V c iH ∗ ( z ) = V b iH ∗ (cid:0) z H , z L , . . . , h iH ( z ) , z iL , . . . , z NH , z NL (cid:1) , with h iH ( z ) = π iH (cid:16) λ iL (1 − z iL ) + (cid:80) j (cid:54) = i ( λ jH (1 − z jH ) + λ jL (1 − z jL )) (cid:17) . Similarly: V c iL ∗ ( z ) = V b iH ∗ (cid:0) z H , z L , . . . , h iH ( z ) , h iL ( z ) , . . . , z NH , z NL (cid:1) , where h iL ( z ) = β ∗ iL (cid:16) λ iL (1 − z iL ) + (cid:80) j (cid:54) = i ( λ jH (1 − z jH ) + λ jL (1 − z jL )) (cid:17) . Note that V c iH ∗ ( · ) = V b iL ∗ ( · ) since there is no switch-over time between Q iH ∗ and Q iL ∗ . There is a switch-overtime between Q iL ∗ and Q ( i +1) H ∗ though: V b ( i +1) H ∗ ( z ) = V c iL ∗ ( z ) σ i N (cid:88) j =1 ( λ jH (1 − z jH ) + λ jL (1 − z jL )) . Now that we can relate V b ( i +1) H ∗ ( · ) to V b iH ∗ ( · ), we can repeat these steps N times to obtain arecursive expression for V b iH ∗ ( · ). This recursive expression is sufficient to compute all momentsof the joint queue length distribution at a visit beginning to Q iH ∗ by differentiation, but theexpression can also be written as an infinite product which converges if and only if ρ <
1. Werefer to [16] for more details.
We define the cycle time C i as the time between two successive visit beginnings to Q i , i =1 , . . . , N . The LST of the distribution of C i , denoted by γ i ( · ), can be expressed in terms5f V b i ( · ) because the type iL customers that are present at the beginning of a visit to Q i are those type iL customers that have arrived during the previous cycle. It is convenient tointroduce the notation (cid:101) V b i ( z iH , z iL ) = V b i (1 , . . . , , z iH , z iL , , . . . , z iH and z iL arethe arguments that correspond respectively to type iH and iL customers. Using this notationwe can write: (cid:101) V b i (1 , z ) = γ i ( λ iL (1 − z )). Hence, the LST of the cycle time distribution is: γ i ( ω ) = (cid:101) V b i (1 , − ωλ iL ) . (4.1)Note that E ( C i ) = E ( S )+ ··· + E ( S N )1 − ρ , which does not depend on i . Higher moments of the cycletime distribution do depend on the cycle starting point.We define the intervisit time I i as the time between a visit completion of Q i and the nextvisit beginning of Q i . The type iH customers present at the beginning of a visit to Q i areexactly those type iH customers that arrived during the previous intervisit time I i . Hence, (cid:101) V b i ( z,
1) = (cid:101) I i ( λ iH (1 − z )), where (cid:101) I ( · ) is the LST of the distribution of I i . This leads to thefollowing expression for the LST of the intervisit time distribution of Q i : (cid:101) I i ( ω ) = (cid:101) V b i (1 − ωλ iH , . (4.2)It is intuitively clear that E ( I i ) = (1 − ρ i ) E ( C ).The LSTs of the distributions of the cycle time and intervisit time are needed later in thispaper. For the visit time of Q i , V i , we mention the LST here for completeness but it will notbe used later: E (e − ωV i ) = (cid:101) V b i ( π iH ( ω ) , β ∗ iL ( ω )) . It is easy to verify that E ( V i ) = ρ i E ( C ). Since high priority customers are served exhaustively, we can use the concept of delay-cycles,sometimes called T -cycles (cf. [21]), introduced by Kella and Yechiali [14] for vacation modelsto find the waiting time LST of a type iH customer, where waiting time is understood asthe time between arrival of a customer into the system and the moment when the customeris taken into service. The waiting time plus service time will be called sojourn time of acustomer. When it comes to computing waiting times in a polling system with priorities, onecan use delay-cycles for any queue that is served exhaustively, cf. [1, 2]. A delay-cycle for atype iH customer is a cycle that starts with a certain initial delay at the moment that thelast type iH customer in the system has been served. In our model this initial delay is eitherthe service of a type iL customer, B iL , or (if no type iL customer is present) an intervisitperiod I i . The delay cycle ends at the first moment after the initial delay when no type iH customer is present in the system again. This is the moment that all type iH customers thathave arrived during the delay, and all of their type iH descendants, have been served. In [1]delay-cycles have been applied to a polling system with two priority levels in an exhaustivelyserved queue. For a type iH customer in the polling model in the present paper, the same6rguments can be used to compute the LST of the waiting time distribution. The fractionof time that the system is in a delay-cycle that starts with the service time B iL of a type iL customer is ρ iL − ρ iH , and the fraction of time that the system is in a delay-cycle that starts withan intervisit period I i , is 1 − ρ iL − ρ iH = − ρ i − ρ iH . We can use the Fuhrmann-Cooper decomposition[11] to obtain the LST of the waiting time distribution of a type iH customer, because fromhis perspective the system is an M/G/ B iL of a type iL customer with probability ρ iL − ρ iH , and an intervisit time I i withprobability − ρ i − ρ iH . This leads to the following expression for the LST of the waiting timedistribution of a type iH customer: E [e − ωW iH ] = (1 − ρ iH ) ωω − λ iH (1 − β iH ( ω )) · (cid:34) ρ iL − ρ iH · − β iL ( ω ) ωE ( B iL ) + 1 − ρ i − ρ iH · − (cid:101) I i ( ω ) ωE ( I i ) (cid:35) . (5.1)Equation (5.1) is similar to the equation found in [1] for high priority customers in an ex-haustive queue. Note that the intervisit time I i is different though, with LST (cid:101) I i ( · ) as definedin Equation (4.2).The GF of the marginal queue length distribution of type iH customers can be found byapplying the distributional form of Little’s Law [13] to the sojourn time distribution: E (cid:0) z N iH (cid:1) = E (cid:16) e − λ iH (1 − z )( W iH + B iH ) (cid:17) . This leads to the following expression: E [ z N iH ] = (1 − ρ iH )(1 − z ) β iH ( λ iH (1 − z )) β iH ( λ iH (1 − z )) − z · (cid:34) ρ iL − ρ iH · − β iL ( λ iH (1 − z ))(1 − z ) λ iH E ( B iL ) + 1 − ρ i − ρ iH · − (cid:101) I i ( λ iH (1 − z ))(1 − z ) λ iH E ( I i ) (cid:35) . (5.2) In this subsection we determine the GF of the marginal queue length distribution of type iL customers, and the LST of the waiting time distribution of type iL customers. In order toobtain these functions, we regard the alternative system with 2 N queues as defined in Section3. The number of type iL customers in the original polling system and their waiting time( excluding the service time) have the same distribution as the number of type iL ∗ customersand their waiting time (again excluding the service time, which is different) in the alternativesystem. From the viewpoint of a type iL ∗ customer, the system is an ordinary polling systemwith synchronised gated service in Q iL ∗ .We apply the Fuhrmann-Cooper decomposition to the alternative polling model with 2 N queues and type iL ∗ customers having completion time B ∗ iL . Using arguments similar as inthe derivation of Equation (3.7) in [3], we find the general form of the GF of the marginalqueue length distribution: E [ z N iL ] = (1 − ρ ∗ iL )(1 − z ) β ∗ iL ( λ iL (1 − z )) β ∗ iL ( λ iL (1 − z )) − z · V c iL (1 , . . . , , z, , . . . , − V b iL (1 , . . . , , z, , . . . , − z )( E ( N ∗ iL | I end ) − E ( N ∗ iL | I begin )) , (5.3)7here ρ ∗ iL = ρ iL − ρ iH and β ∗ iL ( · ) is given by (3.1). Furthermore, N ∗ iL | I end and N ∗ iL | I begin are thenumber of type iL ∗ customers at respectively the visit beginning and visit completion of Q iL ∗ . The visit beginning corresponds to the end of the intervisit period I iL , and the visitcompletion corresponds to the beginning of the intervisit period. Substitution into (5.3) leadsto the following expression: E [ z N iL ] = (1 − ρ iL − ρ iH )(1 − z ) β iL ( λ iL (1 − z ) + λ iH (1 − π iH ( λ iL (1 − z )))) β iL ( λ iL (1 − z ) + λ iH (1 − π iH ( λ iL (1 − z )))) − z · (cid:101) V b i (cid:0) π iH ( λ iL (1 − z )) , β iL ( λ iL (1 − z ) + λ iH (1 − π iH ( λ iL (1 − z )))) (cid:1) − (cid:101) V b i (cid:0) π iH ( λ iL (1 − z )) , z (cid:1) (1 − z ) λ iL (1 − ρ iL − ρ iH ) E ( C ) , (5.4)where we use that E ( N ∗ iL | I end ) − E ( N ∗ iL | I begin ) = λ iL (1 − ρ ∗ iL ) E ( C ) = λ iL (1 − ρ iL − ρ iH ) E ( C ),because this is the mean number of type iL ∗ customers that arrive during the intervisit timeof Q iL ∗ .Applying the distributional form of Little’s Law to (5.4), we obtain the LST of the sojourntime distribution of a type iL customer. Since the sojourn time is W iL + B ∗ iL , the LST of thewaiting time distribution immediately follows: E [e − ωW iL ] = (1 − ρ iL − ρ iH ) ωω − λ iL (1 − β iL ( ω + λ iH (1 − π iH ( ω )))) · (cid:101) V b i (cid:0) π iH ( ω ) , β iL ( ω + λ iH (1 − π iH ( ω ))) (cid:1) − (cid:101) V b i (cid:0) π iH ( ω ) , − ωλ iL (cid:1) ω (1 − ρ iL − ρ iH ) E ( C ) . (5.5) Differentiation of the waiting time LSTs derived in the previous section leads to the followingmean waiting times: E ( W iH ) = ρ iH E ( B iH, res ) + ρ iL E ( B iL, res )1 − ρ iH + 1 − ρ i − ρ iH E ( I i, res ) , (6.1) E ( W iL ) = (cid:18) ρ iL − ρ iH (cid:19) E ( C i, res ) + ρ iH − ρ iH E ( X iH X iL ) λ iL λ iH E ( C ) , (6.2)where B iH, res denotes a residual service time of a type iH customer, with E ( B iH, res ) = E ( B iH )2 E ( B iH ) . We use a similar notation for the residual service time of a type iL customer, theresidual intervisit time, and residual cycle time. Furthermore, X iH and X iL are respectivelythe number of type iH and type iL customers at the beginning of a visit to Q i , so E ( X iH X iL )is obtained by differentiating (cid:101) V b i ( z iH , z iL ) with respect to z iH and z iL (and then setting z iH = z iL = 1).We now present an alternative, direct way to obtain the mean waiting time for a type iL customer by conditioning on the event that an arrival takes place in a visit period, or in an8ntervisit period. E ( W iL ) = E ( V i ) E ( C ) (cid:20) E ( V i, res ) + E ( V i I i ) E ( V i ) + ρ iH − ρ iH E ( V i I i ) E ( V i ) + ρ iL − ρ iH E ( V i, past ) (cid:21) + E ( I i ) E ( C ) (cid:20) E ( I i, res ) + ρ iH − ρ iH ( E ( I i, past ) + E ( I i, res )) + ρ iL − ρ iH (cid:18) E ( V i I i ) E ( I i ) + E ( I i, past ) (cid:19)(cid:21) = 1 E ( C ) (cid:20) E (cid:0) ( V i + I i ) (cid:1) + ρ iL − ρ iH E (cid:0) ( V i + I i ) (cid:1) + ρ iH − ρ iH (cid:0) E ( I i ) + E ( V i I i ) (cid:1)(cid:21) = (cid:18) ρ iL − ρ iH (cid:19) E ( C i )2 E ( C ) + ρ iH − ρ iH (cid:18) E ( I i ) E ( C ) (cid:20) E ( I i )2 E ( I i ) (cid:21) + E ( V i ) E ( C ) E ( I i V i ) E ( V i ) (cid:19) . (6.3)In the above derivation, we use that both the past and residual intervisit time have expectation E ( I i )2 E ( I i ) , and that if a type iL customer arrives during the visit time (with probability E ( V i ) E ( C ) ),the mean length of the following intervisit time equals E ( I i V i ) E ( V i ) . The interpretation of (6.3) isthat a type iL customer always has to wait for the residual cycle time, for the completiontimes of all type iL customers that have arrived during the past cycle time, and for the busyperiods of all type iH customers that have arrived during the intervisit time of the cycle inwhich the type iL customer has arrived.To show that (6.2) and (6.3) are equal, we can rewrite the last term in (6.2): E ( X iH X iL ) = E [( N iL ( V i ) + N iL ( I i )) N iH ( I i )]= E (cid:0) E [( N iL ( V i ) + N iL ( I i )) N iH ( I i )] | I i , V i (cid:1) = E [( λ iL V i + λ iL I i ) λ iH I i ]= λ iL λ iH E ( I i V i ) + λ iL λ iH E ( I i ) , where N j ( T ) denotes the number of type j customers that have arrived during time T ( j = iH, iL ), and V i denotes the length of a visit of the server to Q i . Hence, E ( X iH X iL ) λ iL λ iH E ( C ) = E ( I i V i ) + E ( I i ) E ( C )= E ( I i ) E ( C ) (cid:20) E ( I i )2 E ( I i ) (cid:21) + E ( V i ) E ( C ) E ( I i V i ) E ( V i ) , which coincides with the last term in (6.3). Boxma and Groenendijk [5] have shown that a so-called pseudo-conservation law holds fornonpriority polling systems. We do not discuss this law in detail in the present paper, butwe mention that a generalised version of this law (cf. [18, 9]) holds for systems with multiplepriority levels in each queue: N (cid:88) i =1 K i (cid:88) k =1 ρ ik E ( W ik ) = ρ − ρ N (cid:88) i =1 K i (cid:88) k =1 ρ ik E ( B ik )2 E ( B ik )+ ρ E ( S )2 E ( S ) + (cid:34) ρ − N (cid:88) i =1 ρ i (cid:35) E ( S )2(1 − ρ ) + N (cid:88) i =1 E ( Z ii ) , (7.1)9here S = (cid:80) Ni =1 S i , and K i is the number of priority levels in Q i . In this expression Z ii is the amount of work at Q i when the server leaves this queue and depends on the servicediscipline. It is well-known that for gated service, E ( Z ii ) = ρ i E ( C ) and for exhaustiveservice, E ( Z ii ) = 0. The pseudo-conservation law also holds for polling systems with mixedgated/exhaustive service in some or all of the queues. If Q i receives mixed gated/exhaustiveservice, we have K i = 2, and E ( Z ii ) = ρ iL ρ i E ( C ). Example 1
In order to illustrate the effect of using a mixed gated/exhaustive service discipline in apolling system with priorities, we compare it to the commonly used gated and exhaustiveservice disciplines. In this example we use a polling system which consists of two queues, Q and Q . Customers in Q are divided into high priority customers, arriving with arrival rate λ H = , and low priority customers, with arrival rate λ L = . Customers in Q all havethe same priority level and arrive with arrival rate λ = . All service times are exponentiallydistributed with mean 1. The switch-over times S and S are also exponentially distributedwith mean 1, which results in a mean cycle time of E ( C ) = 10. The service discipline in Q isgated, the service discipline in Q is varied: gated, exhaustive and mixed gated/exhaustive.Results for a queue with two priority levels and purely gated or exhaustive service are obtainedin [1].Table 1 displays the mean and the variance of the waiting times of the three customer typesunder the three service disciplines. We conclude that the mixed gated/exhaustive service isa major improvement for the high priority customers in Q , whereas the mean waiting timesof the low priority customers in Q and the customers in Q hardly deteriorate. Of coursein systems where ρ H is quite high, the negative impact can be bigger and one has to decideexactly how far one wants to go in giving extra advantages to customers that already receivehigh priority. When comparing the mixed gated/exhaustive strategy to a system with purelyexhaustive service in Q , we conclude that the improvement is not so much in the meanwaiting time for high priority customers, but mostly in the mean and variance of the waitingtime for customers in Q . Gated Exhaustive Mixed G/E E ( W H ) 9.578 2.520 2.338 E ( W L ) 14.366 6.300 14.575 E ( W ) 9.690 14.880 10.513Var( W H ) 56.739 9.290 6.496Var( W L ) 101.616 32.812 118.217Var( W ) 58.513 231.256 76.371Table 1: Numerical results for Example 1. The switch-over times S and S are exponentiallydistributed with mean 1. The mixed gated/exhaustive service discipline is compared to gatedand exhaustive service. 10ated Exhaustive Mixed G/E E ( W H ) 63.187 11.333 11.167 E ( W L ) 94.781 28.333 90.417 E ( W ) 63.251 68.000 64.000Var( W H ) 847.377 195.508 183.907Var( W L ) 894.173 315.823 850.199Var( W ) 853.777 1386.100 928.914Table 2: Numerical results for Example 1. Switch-over times are deterministic: S = S = 10.It is noteworthy that the mixed gated/exhaustive service discipline does not always have anegative effect on the mean waiting time of low priority customers in Q , E ( W L ), compared tothe gated service discipline. If, for example, the switch-over times are taken to be deterministicwith value 10, the mean waiting time for low priority customers is significantly less for themixed gated/exhaustive service than for gated service, as can be seen in Table 2. Comparedto gated service, type 1 H customers benefit strongly from the mixed gated/exhaustive servicediscipline, and even type 1 L customers benefit from it. The mean waiting time for customersin Q has increased, but only marginally.In order to get more understanding of this surprising behaviour of the waiting time of lowpriority customers as function of the arrival intensities λ H and λ L , we use a simplifiedmodel which leads to more insightful expressions, but displays the same characteristics as themodel that was analysed in the previous paragraph. Instead of analysing a polling model,we analyse an M/G/ Q touse familiar notation, contains high (type 1 H ) and low (type 1 L ) priority customers. Alsohere high priority customers are served before low priority customers. The service timesof both customers types are exponentially distributed with mean 1. This is for notationalreasons only, for this example we actually only require that both service times are identicallydistributed. One server vacation has a fixed length S . If the server does not find any customerswaiting upon arrival from a vacation, he takes another vacation of length S and so on. Inorder to stay consistent with the notation used earlier, we denote the occupation rate ofhigh and low priority customers by respectively ρ H and ρ L . The total occupation rate is ρ = ρ = ρ H + ρ L . Note that in this example λ H = ρ H and λ L = ρ L . We now comparethe mean waiting times of type 1 L customers in the system with purely gated service andthe system with mixed gated/exhaustive service. For this simplified model, we can writedown explicit expressions that have been obtained by differentiating the LSTs and solvingthe resulting equations. These expressions could also have been obtained by using MeanValue Analysis (MVA) for polling systems [22, 24].Gated service: E ( W L ) = (1 + ρ + ρ H ) (cid:18) S − ρ ) + ρ − ρ (cid:19) , (8.1)Mixed G/E service: E ( W L ) = ρ (1 − ρ )(1 − ρ H ) + S (1 + ρ (1 − ρ H ))2(1 − ρ )(1 − ρ H ) . (8.2)Now we analyse the behaviour of these waiting times as we vary λ H between 0 and ρ , whilekeeping λ H + λ L = ρ constant. Substitution of λ H = 0 shows that the mean waiting times11n the gated and mixed gated/exhaustive system are equal: E ( W L | ρ H = 0) = S (1 + ρ )2(1 − ρ ) + ρ − ρ . Letting λ H → ρ leads to the following expressions:Gated service: E ( W L | ρ H → ρ ) = ρ (1 + 2 ρ )1 − ρ + S (1 + 2 ρ )2(1 − ρ ) ,Mixed G/E service: E ( W L | ρ H → ρ ) = ρ (1 − ρ ) + S (1 + 2 ρ )2(1 − ρ ) .Two interesting things can be concluded from these two equations for the case λ H → ρ : • for fixed ρ , E ( W L ) in a gated system is always less than E ( W L ) in a mixed gated/exhaustivesystem, • the difference between E ( W L ) in a gated system and E ( W L ) in a mixed gated/exhaustivesystem does not depend on S .Focussing on the mean waiting time of type 1 L customers only, we conclude that a gatedsystem performs the same as a mixed gated/exhaustive system as ρ L = ρ , and that a gatedsystem always performs better when ρ L →
0. For 0 < ρ L < ρ the vacation time S determineswhich system performs better. By taking derivatives of (8.1) and (8.2) with respect to ρ H and letting ρ H →
0, one finds that the mean waiting time of a type 1 L customer in amixed gated/exhaustive system is less than in a purely gated system when ρ H →
0, if andonly if
S > ρ ρ . Since a gated system always outperforms a mixed gated/exhaustive systemwhen λ H → ρ , for S > ρ ρ there must be (at least) one value of λ H for which the twosystems perform the same. Further inspection of the derivatives gives the insight that in agated system the relation between E ( W L ) and λ H is a straight line, which can also be seenimmediately from Equation (8.1). In a mixed gated/exhaustive system, the relation between E ( W L ) and λ H is not a straight line, both the first and second derivative with respect to λ H are strictly positive. This means that for S ≤ ρ ρ the gated system always performsbetter than the mixed gated/exhaustive system for any value of λ H >
0, and for
S > ρ ρ themixed gated/exhaustive system performs better than the gated system for 0 < λ H < λ ∗ H .The value of λ ∗ H can be determined analytically: λ ∗ H = ρ S − ρ ρ S + ρ ρ . From this expression we conclude that lim S →∞ λ ∗ H = ρ . Although we have studied only thevacation model, the conclusions are also valid for more general settings, like polling modelswith non-deterministic switch-over times, but the expressions are by far not as appealing.We visualise the findings of the present section in Figure 1, where we show three plots ofthe mean waiting time of type 1 L customers against λ H . The model considered is the sameas in the beginning of the present section (two queues, gated service in Q ) except for theswitch-over times S and S , which are now deterministic. We compare gated service in Q to mixed gated/exhaustive service for three different switch-over times (notice that the scalesof the three plots in Figure 1 are different). 12 .1 0.2 0.3 0.4 0.5 0.6 GatedMixed G (cid:144) E Λ H E H W L L a) S = 1 Λ H E H W L L b) S = 10 Λ H ´ ´ ´ ´ ´ ´ ´ E H W L L c) S = 10 Figure 1: Mean waiting time of type 1 L customers in the polling model discussed in Example1. For gated and mixed gated/exhaustive service E ( W L ) is plotted against λ H while keeping λ L + λ H constant. The switch-over times S = S = S/ Example 2
In the previous example we showed that the mixed gated/exhaustive service discipline doesnot necessarily have a negative impact on the mean waiting times of low priority customers. Inthis example we aim at giving a better comparison of the performance of the gated, exhaustiveand mixed gated/exhaustive service disciplines in a polling system with priorities. The pollingsystem considered consists of two queues, each having high and low priority customers. Theswitch-over times S and S are exponentially distributed with mean 10. Service times of allcustomer types are exponentially distributed with mean 1. The arrival rates of the variouscustomer types are: λ H = λ L = , and λ H = λ L = . The total occupation rate of thispolling system is ρ = , and we deliberately choose a system where the occupation rates ofthe two queues are very different, and the switch-over times are relatively high compared tothe service times. The reason is that we envision production systems as the main applicationfor the present paper (see also Section 1). In these applications large setup times are verycommon (see, e.g., [23]).Table 3 shows the mean and variance of the waiting times of all customer types of this pollingsystem for all combinations of gated, exhaustive and mixed gated/exhaustive service. Weleave it up to the reader to pick his favourite combination of service disciplines, but ourpreference goes out to the system with exhaustive service in Q and mixed gated/exhaustiveservice in Q because in our opinion the best combination of low mean waiting times andmoderate variances is obtained in this system. Many extensions or variations of the model discussed in the present paper can be thought of.In this section we discuss some of them.
A globally gated system.
The globally gated service discipline has received quite someattention in polling systems. Instead of setting the gates at the beginning of a visit to13ueue Service discipline E ( W iL ) E ( W iH ) Var( W iL ) Var( W iH )1 gated 141.81 119.99 5166.03 4660.092 gated 222.95 146.82 5917.70 3560.67Queue Service discipline E ( W iL ) E ( W iH ) Var( W iL ) Var( W iH )1 gated 165.49 140.03 11087.40 9411.432 exhaustive 59.45 17.83 1862.57 651.03Queue Service discipline E ( W iL ) E ( W iH ) Var( W iL ) Var( W iH )1 gated 147.38 124.71 6406.11 5658.442 gatedexhaustive 209.86 16.98 6213.92 555.67Queue Service discipline E ( W iL ) E ( W iH ) Var( W iL ) Var( W iH )1 exhaustive 97.63 78.10 4252.19 3784.992 gated 224.00 147.51 6186.88 3690.81Queue Service discipline E ( W iL ) E ( W iH ) Var( W iL ) Var( W iH )1 exhaustive 119.80 95.84 9516.58 7952.092 exhaustive 61.62 18.49 2136.19 728.97Queue Service discipline E ( W iL ) E ( W iH ) Var( W iL ) Var( W iH )1 exhaustive 102.18 81.75 5193.21 4533.582 gatedexhaustive 211.90 17.27 6722.53 586.84Queue Service discipline E ( W iL ) E ( W iH ) Var( W iL ) Var( W iH )1 gatedexhaustive 140.95 77.96 5140.20 3756.122 gated 223.45 147.15 6045.55 3622.49Queue Service discipline E ( W iL ) E ( W iH ) Var( W iL ) Var( W iH )1 gatedexhaustive 166.85 94.38 11655.90 7574.672 exhaustive 60.39 18.12 1978.87 684.25Queue Service discipline E ( W iL ) E ( W iH ) Var( W iL ) Var( W iH )1 gatedexhaustive 146.87 81.41 6452.48 4462.042 gatedexhaustive 210.82 17.10 6451.10 569.08Table 3: Expectation and variance of the waiting times of the polling model discussed inSection 8, Example 2.a certain queue, the globally gated service discipline states that all gates are set at thebeginning of a cycle, which is the start of a visit to an arbitrarily chosen queue. The modelunder consideration can be analysed using similar techniques if high priority customers areserved exhaustively, but low priority customers are served according to the globally gatedservice discipline. One would first have to build a similar model that contains 2 N queues14nd determine the joint queue length distribution at visit beginnings and endings. Thecycle time, starting at the moment that all gates are set, can be expressed in terms of theGF of the number of customers at the beginning of that cycle. Waiting times for highpriority customers can be obtained using delay-cycles again, and waiting times for low prioritycustomers can be obtained using the Fuhrmann-Cooper decomposition. The LST of thewaiting time distribution of low priority customers gets more complicated as the queue getsserved later in the cycle. More than two priority levels.
It is possible to analyse a similar model as the one ofSection 2, but with more than two, say K i , priority levels in Q i . These K i priority levels stillhave to be divided into two categories: high priority levels 1 , . . . , k i that receive exhaustiveservice, and low priority levels k i + 1 , . . . , K i that receive gated service. The methodologyfrom Section 5 can be used, combined with the techniques that are used to analyse a pollingmodel with multiple priority levels, cf. [2]. A mixture of gated and exhaustive without priorities.
One could think of a systemwhere each queue contains two customer classes having respectively the exhaustive and gatedservice discipline, but service is First-Come-First-Served (FCFS). The model is similar to themodel discussed in this paper, with the exception that no “overtaking” takes place. Customersthat are served exhaustively will not be served before any “gated customers” standing in frontof this gate, but they are allowed to pass the gate. The joint queue length distributions atpolling epochs and the cycle times are the same as for the system considered in the presentpaper. Since no overtaking takes place, the waiting times can be found without the use of delaycycles. Nevertheless, analysis of the waiting times is quite tedious because a visit of a serverto Q i consists of three parts. The third part is the service of exhaustive customers behind thegate, the first part is the service of the gated customers that have arrived during the “previousthird part” and the second part is the FCFS service of both gated and exhaustive customersthat have arrived during the previous intervisit time of Q i . A combination of this non-prioritymixture of gated and exhaustive, and the service discipline discussed in the present paper isdiscussed by Fiems et al. [8]. They introduce, albeit in the different setting of a vacationqueue modelled in discrete time, a service discipline where high priority customers in front ofthe gate are served before low priority customers waiting in front of the gate. The differencewith the model discussed in the present paper, is that high priority customers entering thequeue while it is being visited can pass the gate, but are not allowed to overtake low prioritycustomers standing in front of the gate. Acknowledgements
The authors wish to thank Erik Winands for his many helpful remarks and discussions. Hiscontribution to the present paper is very much appreciated. Our gratitude also goes out toJacques Resing who suggested the mixed gated/exhaustive service discipline. Finally, theauthors thank Onno Boxma for valuable discussions and for useful comments on earlier draftsof the present paper. 15 eferences [1] M. A. A. Boon, I. J. B. F. Adan, and O. J. Boxma. A two-queue polling model with twopriority levels in the first queue.
ValueTools 2008 (Third International Conference onPerformance Evaluation Methodologies and Tools, Athens, Greece, October 20-24, 2008) .[2] M. A. A. Boon, I. J. B. F. Adan, and O. J. Boxma. A polling model with multiplepriority levels.
Eurandom report 2008-029,
Eurandom , 2008.[3] S. C. Borst.
Polling Systems , volume 115 of
CWI Tracts . 1996.[4] O. J. Boxma. Polling systems.
From universal morphisms to megabytes: A Baayen spaceodyssey. Liber amicorum for P.C. Baayen. CWI, Amsterdam , pages 215–230, 1994.[5] O. J. Boxma and W. P. Groenendijk. Pseudo-conservation laws in cyclic-service systems.
Journal of Applied Probability , 24(4):949–964, 1987.[6] O. J. Boxma and J. A. Weststrate. Waiting times in polling systems with Markovianserver routing. In
Messung, Modellierung und Bewertung von Rechensystemen und Net-zen , eds. G. Stiege and J. S. Lie, pages 89–105. Springer Verlag, Berlin, 1989.[7] J. W. Cohen.
The Single Server Queue . North-Holland, Amsterdam, revised edition,1982.[8] D. Fiems, S. De Vuyst, and H. Bruneel. The combined gated-exhaustive vacation systemin discrete time.
Performance Evaluation , 49:227–239, 2002.[9] L. Fournier and Z. Rosberg. Expected waiting times in polling systems under prioritydisciplines.
Queueing Systems , 9(4):419–439, 1991.[10] S. W. Fuhrmann. Performance analysis of a class of cyclic schedules. Technical memo-randum 81-59531-1, Bell Laboratories, March 1981.[11] S. W. Fuhrmann and R. B. Cooper. Stochastic decompositions in the
M/G/
Operations Research , 33(5):1117–1129, 1985.[12] J. Gianini and D. R. Manfield. An analysis of symmetric polling systems with two priorityclasses.
Performance Evaluation , 8:93–115, 1988.[13] J. Keilson and L. D. Servi. The distributional form of Little’s Law and the Fuhrmann-Cooper decomposition.
Operations Research Letters , 9(4):239–247, 1990.[14] O. Kella and U. Yechiali. Priorities in
M/G/
NavalResearch Logistics , 35:23–34, 1988.[15] A. Khamisy, E. Altman, and M. Sidi. Polling systems with synchronization constraints.
Annals of Operations Research , 35:231 – 267, 1992.[16] J. A. C. Resing. Polling systems and multitype branching processes.
Queueing Systems ,13:409 – 426, 1993. 1617] M. Sharafali, H. C. Co, and M. Goh. Production scheduling in a flexible manufacturingsystem under random demand.
European Journal of Operational Research , 158:89 – 102,2004.[18] S. Shimogawa and Y. Takahashi. A pseudo-conservation law in a cyclic-service systemwith priority classes.
IEICE Research Report , (IN88-86):13–18, 1988.[19] M. M. Srinivasan. Non-deterministic polling systems.
Management Science , 37:667–681,1991.[20] H. Takagi. Priority queues with setup times.
Operations Research , 38(4):667–677, 1990.[21] H. Takagi.
Queueing Analysis: A Foundation Of Performance Evaluation , volume 1:Vacation and Priority Systems, Part 1. North-Holland, Amsterdam, 1991.[22] A. Wierman, E. M. M. Winands, and O. J. Boxma. Scheduling in polling systems.
Performance Evaluation , 64:1009–1028, 2007.[23] E. M. M. Winands.
Polling, Production & Priorities . PhD thesis, Eindhoven Universityof Technology, 2007.[24] E. M. M. Winands, I. J. B. F. Adan, and G.-J. van Houtum. Mean value analysis forpolling systems.