A Two-Queue Polling Model with Two Priority Levels in the First Queue
AA Two-Queue Polling Model with Two Priority Levels in the First Queue ∗† M.A.A. Boon ‡ [email protected] I.J.B.F. Adan † [email protected] O.J. Boxma † [email protected], 2008 Abstract
In this paper we consider a single-server cyclic polling system consisting of twoqueues. Between visits to successive queues, the server is delayed by a random switch-over time. Two types of customers arrive at the first queue: high and low prioritycustomers. For this situation the following service disciplines are considered: gated,globally gated, and exhaustive. We study the cycle time distribution, the waitingtimes for each customer type, the joint queue length distribution at polling epochs,and the steady-state marginal queue length distributions for each customer type.
Keywords:
Polling, priority levels, queue lengths, waiting times
A polling model is a single-server system in which the server visits n queues Q , . . . , Q n incyclic order. Customers that arrive at Q i are referred to as type i customers. The specialfeature of the model considered in the present paper is that, within a customer type, wedistinguish high and low priority customers. More specifically, we study a polling systemwhich consists of two queues, Q and Q . The first of these queues contains customersof two priority classes, high ( H ) and low ( L ). The exhaustive, gated and globally gatedservice disciplines are studied. ∗ The research was done in the framework of the BSIK/BRICKS project, and of the European Networkof Excellence Euro-FGI. † The present paper is an adapted and extended version of [3]. ‡ Eurandom and Department of Mathematics and Computer Science, Eindhoven University of Tech-nology, P.O. Box 513, 5600MB Eindhoven, The Netherlands a r X i v : . [ m a t h . P R ] A ug ur motivation to study a polling model with priorities is that the performance of apolling system can be improved through the introduction of priorities. In productionenvironments, e.g., one could give highest priority to jobs with a service requirement belowa certain threshold level. This might decrease the mean waiting time of an arbitrarycustomer without having to purchase additional resources [24]. Priority polling modelsalso can be used to study traffic intersections where conflicting traffic flows face a greenlight simultaneously; e.g. traffic which takes a left turn may have to give right of way toconflicting traffic that moves straight on, even if the traffic light is green for both trafficflows. Another application is discussed in [9], where a priority polling model is used to studyscheduling of surgery procedures in medical emergency rooms. In the computer sciencecommunity the Bluetooth and 802.11 protocols are frequently modelled as polling systems,cf. [17, 18, 19, 27]. Many scheduling policies that have been considered or implemented inthese protocols involve different priority levels in order to improve Quality-of-Service (QoS)for traffic that is very sensitive to delays or loss of data, such as Voice over Wireless IP.The 802.11e amendment defines a set of QoS enhancements for wireless LAN applicationsby differentiating between high priority traffic, like streaming multimedia, and low prioritytraffic, like web browsing and email traffic.Although there is quite an extensive amount of literature available on polling systems, onlyvery few papers treat priorities in polling models. Most of these papers only provide ap-proximations or focus on pseudo-conservation laws. In [24] exact mean waiting time resultsare obtained using the Mean Value Analysis (MVA) framework for polling systems, devel-oped in [26]. The MVA framework can only be used to find the first moment of the waitingtime distribution for each customer type, and the mean residual cycle time. The maincontribution of the present paper is the derivation of Laplace Stieltjes Transforms (LSTs)of the distributions of the marginal waiting times for each customer type; in particular itturns out to be possible to obtain exact expressions for the waiting time distributions ofboth high and low priority customers at a queue of a polling system. Probability Generat-ing Functions (PGFs) are derived for the joint queue length distribution at polling epochs,and for the steady-state marginal queue length distribution of the number of customers atan arbitrary epoch.The present paper is structured as follows: Section 2 gathers known results of nonprioritypolling models which are relevant for the present study. Sections 3 (gated), 4 (globallygated), and 5 (exhaustive) give new results on the priority polling model. In each of thesections we successively discuss the joint queue length distribution at polling epochs, thecycle time distribution, the marginal queue length distributions and waiting time distribu-tions. The mean waiting times are given at the end of each section. A numerical exampleis presented in Section 6 to illustrate some of the improvements that can be obtained byintroducing prioritisation in a polling system.2 Notation and description of the nonpriority pollingmodel
The model that is considered in this section, is a nonpriority polling model with two queues( Q and Q ). We consider three service disciplines: gated, globally gated, and exhaustive.The gated service discipline states that during a visit to Q i , the server serves only thosetype i customers who are present at the polling epoch. All type i customers that arriveduring this visit will be served in the next cycle. In this respect, a cycle is the time betweentwo successive visit beginnings to a queue. The exhaustive service discipline states thatwhen the server arrives at Q i , all type i customers are served until no type i customer ispresent in the system. We also consider the globally gated service discipline, which meansthat during a cycle only those customers will be served that were present at the beginningof that cycle.Customers of type i arrive at Q i according to a Poisson process with arrival rate λ i ( i =1 , . Service times can follow any distribution, and we assume that a customer’s servicetime is independent of other service times and independent of the arrival processes. TheLST of the distribution of the generic service time B i of type i customers is denoted by β i ( · ) .The fraction of time that the server is serving customers of type i equals ρ i := λ i E ( B i ) .Switches of the server from Q i to Q i +1 (all indices modulo 2), require a switch-over time S i .The LST of this switch-over time distribution is denoted by σ i ( · ) . The fraction of time thatthe server is working (i.e., not switching) is ρ := ρ + ρ . We assume that ρ < , which isa necessary and sufficient condition for the steady state distributions of cycle times, queuelengths and waiting times to exist.[22] studied this model, but without switch-over times and only with the exhaustive servicediscipline. [11] analysed this polling system for any number of queues, and for both gatedand exhaustive service disciplines. [12] obtained results for a polling system with switch-over times (but only exhaustive service) by relating the PGFs of the joint queue lengthdistributions at visit beginnings, visit endings, service beginnings and service endings. [20]was the first to point out the relation between polling systems and Multitype BranchingProcesses with immigration in each state. His results can be applied to polling models inwhich each queue satisfies the following property: 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 canbe any n -dimensional probability generating function.We use this property, and the relation to Multitype Branching Processes, to find results forour polling system with two queues, two priorities in the first queue, and gated, globallygated, and exhaustive service discipline. Notice that, unlike the gated and exhaustiveservice disciplines, the globally gated service discipline does not satisfy Property 2.1. Butthe results obtained by Resing also hold for a more general class of polling systems, namely3hose which satisfy the following (weaker) property that is formulated in [4]: Property 2.2
If there are k i customers present at Q i at the beginning (or the end) of avisit to Q π ( i ) , with π ( i ) ∈ { , . . . , n } , then during the course of the visit to Q i , each of these k i customers will effectively be replaced in an i.i.d. manner by a random population havingprobability generating function h i ( z , . . . , z n ) , which can be any n -dimensional probabilitygenerating function.Globally gated and gated are special cases of the synchronised gated service discipline,which states that only customers in Q i will be served that were present at the momentthat the server reaches the “parent queue” of Q i : Q π ( i ) . For gated service, π ( i ) = i , forglobally gated service, π ( i ) = 1 . The synchronised gated service discipline is discussed in[16], but no observation is made that this discipline is a member of the class of pollingsystems satisfying Property 2.2 which means that results as obtained in [20] can be extendedto this model.[5] combined the results of [20] and [12] to find a relation between the PGFs of the marginalqueue length distribution for polling systems with and without switch-over times, expressedin the Fuhrmann-Cooper queue length decomposition form [13]. The probability generating function h i ( z , . . . , z n ) which is mentioned in Property 2.1 de-pends on the service discipline. In a polling system with two queues and gated service wehave h i ( z , z ) = β i ( λ (1 − z ) + λ (1 − z )) . For exhaustive service this PGF becomes h i ( z , z ) = π i ( (cid:80) j (cid:54) = i λ j (1 − z j )) , where π i ( · ) is the LST of a busy period (BP) distribu-tion in an M/G/ system with only type i customers, so it is the root of the equation π i ( ω ) = β i ( ω + λ i (1 − π i ( ω ))) . We choose the beginning of a visit to Q as start of acycle. In order to find the joint queue length distribution at the beginning of a cycle, werelate the numbers of customers in each queue at the beginning of a cycle to those at thebeginning of the previous cycle. Customers always enter the system during a switch-overtime, or during a visit period. The first group is called immigration , whereas a customerfrom the second group is called offspring of the customer that is served at the moment ofhis arrival. We define the immigration PGF for each switch-over time and the offspringPGF for each visit period analogous to [20]. The immigration PGFs are: g (2) ( z , z ) = σ ( λ (1 − z ) + λ (1 − z )) ,g (1) ( z , z ) = σ ( λ (1 − z ) + λ (1 − h ( z , z ))) .g (2) ( z , z ) is the PGF of the joint distribution of type and customers that arrive during S . For S things are slightly more complicated, since type customers arriving during S may be served before the end of the cycle, and generate offspring. g (1) ( z , z ) is the jointPGF of the type and customers present at the end of the cycle that either arrived during4 , or are offspring of type 2 customers that arrived during S . The total immigration PGFis the product of these two PGFs: g ( z , z ) = (cid:89) i =1 g ( i ) ( z , z ) = g (1) ( z , z ) g (2) ( z , z ) . We define the offspring PGFs for each visit period in a similar manner: f (2) ( z , z ) = h ( z , z ) ,f (1) ( z , z ) = h ( z , h ( z , z )) . The term for Q is again slightly more complicated than the term for Q , since type 2customers arriving during a server visit to Q may be served before the end of the cycle,and generate offspring.[20] shows that the following recursive expression holds for the joint queue length PGF atthe beginning of a cycle (starting with a visit to Q ): P ( z , z ) = g ( z , z ) P (cid:0) f (1) ( z , z ) , f (2) ( z , z ) (cid:1) . This expression can be used to compute moments of the joint queue length distribution.Alternatively, iteration of this expression yields the following closed form expression for P ( z , z ) : P ( z , z ) = ∞ (cid:89) n =0 g ( f n ( z , z )) , (2.1)where we use the following recursive definition for f n ( z , z ) , n = 0 , , , . . . : f n ( z , z ) = ( f (1) ( f n − ( z , z )) , f (2) ( f n − ( z , z ))) ,f ( z , z ) = ( z , z ) . [20] proves that this infinite product converges if and only if ρ < .We can relate the joint queue length distribution at other polling epochs to P ( z , z ) . Wedenote the PGF of the joint queue length distribution at a visit beginning to Q i by V b i ( · ) ,so P ( · ) = V b ( · ) . The PGF of the joint queue length distribution at a visit completion to Q i is denoted by V c i ( · ) . The following relations hold: V b ( z , z ) = V c ( z , z ) σ ( λ (1 − z ) + λ (1 − z ))= V b ( z , h ( z , z )) σ ( λ (1 − z ) + λ (1 − z ))= V b ( z , f (2) ( z , z )) g (2) ( z , z ) , (2.2) V b ( z , z ) = V c ( z , z ) σ ( λ (1 − z ) + λ (1 − z ))= V b ( h ( z , z ) , z ) σ ( λ (1 − z ) + λ (1 − z )) . (2.3)5 .2 Cycle time The cycle time, starting at a visit beginning to Q , is the sum of the visit times to Q and Q , and the two switch-over times which are independent of the visit times. Since type 2customers who arrive during the visit to Q or the switch from Q to Q will be servedduring the visit to Q , it can be shown that the LST of the distribution of the cycle time C , γ ( · ) , is related to P ( · ) as follows: γ ( ω ) = σ ( ω + λ (1 − φ ( ω ))) σ ( ω ) P ( φ ( ω + λ (1 − φ ( ω ))) , φ ( ω )) , (2.4)where φ i ( · ) is the LST of the distribution of the time that the server spends at Q i due tothe presence of one type i customer there. For gated service φ i ( · ) = β i ( · ) , for exhaustiveservice φ i ( · ) = π i ( · ) . A proof of (2.4) can be found in [8].In some cases it is convenient to choose a different starting point for a cycle, for examplewhen analysing a polling system with exhaustive service. If we define C ∗ to be the timebetween two successive visit completions to Q , the LST of its distribution, γ ∗ ( · ) , is: γ ∗ ( ω ) = σ ( ω + λ (1 − φ ( ω )) + λ (1 − φ ( ω + λ (1 − φ ( ω ))))) · σ ( ω + λ (1 − φ ( ω ))) V c ( φ ( ω ) , φ ( ω + λ (1 − φ ( ω )))) , (2.5)with V c ( z , z ) = P ( h ( z , z ) , z ) . We denote the PGF of the steady-state marginal queue length distribution of Q at thevisit beginning by (cid:101) V b ( z ) = V b ( z, . Analogously we define (cid:101) V b ( · ) , (cid:101) V c ( · ) , and (cid:101) V c ( · ) . It isshown in [5] that the steady-state marginal queue length of Q i can be decomposed into twoparts: the queue length of the corresponding M/G/ queue with only type i customers,and the queue length at an arbitrary epoch during the intervisit period of Q i , denoted by N i | I . [5] show that by virtue of PASTA, N i | I has the same distribution as the number oftype i customers seen by an arbitrary type i customer arriving during an intervisit period,which equals E ( z N i | I ) = E ( z N i | I begin ) − E ( z N i | I end )(1 − z )( E ( N i | I end ) − E ( N i | I begin )) , where N i | I begin is the number of type i customers at the beginning of an intervisit period I i ,and N i | I end is the number of type i customers at the end of I i . Since the beginning of anintervisit period coincides with the completion of a visit to Q i , and the end of an intervisitperiod coincides with the beginning of a visit, we know the PGFs for the distributions ofthese random variables: (cid:101) V c i ( · ) and (cid:101) V b i ( · ) . This leads to the following expression for thePGF of the steady-state queue length distribution of Q i at an arbitrary epoch, E [ z N i ] : E [ z N i ] = (1 − ρ i )(1 − z ) β i ( λ i (1 − z )) β i ( λ i (1 − z )) − z · (cid:101) V c i ( z ) − (cid:101) V b i ( z )(1 − z )( E ( N i | I end ) − E ( N i | I begin )) . (2.6)614] show that the distributional form of Little’s law can be used to find the LST ofthe marginal waiting time distribution: E ( z N i ) = E ( e − λ i (1 − z )( W i + B i ) ) , hence E ( e − ωW i ) = E [(1 − ωλ i ) N i ] /β i ( ω ) . This can be substituted into (2.6): E [ e − ωW i ] = (1 − ρ i ) ωω − λ i (1 − β i ( ω )) · (cid:101) V c i (cid:16) − ωλ i (cid:17) − (cid:101) V b i (cid:16) − ωλ i (cid:17) ( E ( N i | I end ) − E ( N i | I begin )) ω/λ i = E [ e − ωW i | M/G/ ] E (cid:34)(cid:18) − ωλ i (cid:19) N i | I (cid:35) . (2.7)The interpretation of this formula is that the waiting time of a type i customer in a pollingmodel is the sum of two independent random variables: the waiting time of a customer inan M/G/ queue with only type i customers, W i | M/G/ , and the remaining intervisit timefor a customer that arrives at an arbitrary epoch during the intervisit time of Q i .For gated service, the number of type i customers at the beginning of a visit to Q i isexactly the number of type i customers that arrived during the previous cycle, startingat Q i . In terms of PGFs: (cid:101) V b i ( z ) = γ i ( λ i (1 − z )) . The number of type i customers at theend of a visit to Q i are exactly those type i customers that arrived during this visit. Interms of PGFs: (cid:101) V c i ( z ) = γ i ( λ i (1 − β i ( λ i (1 − z )))) . We can rewrite E ( N i | I end ) − E ( N i | I begin ) as λ i E ( I i ) , because this is the number of type i customers that arrive during an intervisittime. In Section 2.4 we show that λ i E ( I i ) = λ i (1 − ρ i ) E ( C ) . Using these expressions wecan rewrite Equation (2.7) for gated service to: E [ e − ωW i ] = (1 − ρ i ) ωω − λ i (1 − β i ( ω )) · γ i ( λ i (1 − β i ( ω ))) − γ i ( ω )(1 − ρ i ) ωE ( C ) . (2.8)For exhaustive service, (cid:101) V c i ( z ) = 1 , because Q i is empty at the end of a visit to Q i . Thenumber of type i customers at the beginning of a visit to Q i in an exhaustive polling systemis equal to the number of type i customers that arrived during the previous intervisit timeof Q i . Hence, (cid:101) V b i ( z ) = (cid:101) I i ( λ i (1 − z )) , where (cid:101) I i ( · ) is the LST of the intervisit time distributionfor Q i . Substitution of (cid:101) I i ( ω ) = (cid:101) V b i (1 − ωλ i ) in (2.7) leads to the following expression for theLST of the steady-state waiting time distribution of a type i customer in an exhaustivepolling system: E [ e − ωW i ] = (1 − ρ i ) ωω − λ i (1 − β i ( ω )) · − (cid:101) I i ( ω ) ωE ( I i ) . (2.9)To the best of our knowledge, the following result is new. Proposition 2.3
Let the cycle time C ∗ i be the time between two successive visit com-pletions to Q i . The LST of the cycle time distribution is given by (2.5). An equivalentexpression for E [ e − ωW i ] if Q i is served exhaustively, is: E [ e − ωW i ] = 1 − γ ∗ i ( ω − λ i (1 − β i ( ω )))( ω − λ i (1 − β i ( ω ))) E ( C ) (2.10) = E [ e − ( ω − λ i (1 − β i ( ω ))) C ∗ i, res ] , C ∗ i, res is the residual length of C ∗ i . Proof:
The cycle time is the length of an intervisit period I i plus the length of a visit V i , which isthe time required to serve all type i customers that have arrived during I i , and their type i descendants. Hence, the following equation holds: γ ∗ i ( ω ) = (cid:101) I i ( ω + λ i (1 − π i ( ω ))) . (2.11)We use this equation to find the inverse relation: (cid:101) I i ( ω + λ i (1 − π i ( ω ))) = γ ∗ i ( ω )= γ ∗ i ( ω + λ i (1 − π i ( ω )) − λ i (1 − π i ( ω )))= γ ∗ i ( ω + λ i (1 − π i ( ω )) − λ i (1 − β i ( ω + λ i (1 − π i ( ω ))))) . If we substitute s := ω + λ i (1 − π i ( ω )) , we find (cid:101) I i ( s ) = γ ∗ i ( s − λ i (1 − β i ( s ))) . (2.12)Substitution of (2.12) into (2.9) gives (2.10). (cid:3) Remark 2.4
We can write (2.11) and (2.12) as follows: γ ∗ i ( ω ) = (cid:101) I i ( ψ ( ω )) , (cid:101) I i ( s ) = γ ∗ i ( φ ( s )) , where φ ( · ) equals the Laplace exponent of the Lévy process (cid:80) N ( t ) j =1 B i,j − t , with N ( t ) aPoisson process with intensity λ i , and with ψ ( ω ) = ω + λ i (1 − π i ( ω )) , which is known tobe the inverse of φ ( · ) . The focus of this paper is on LST and PGF of distribution functions, not on their mo-ments. Moments can be obtained by differentiation, and are also discussed in [24]. In thissubsection we will only mention some results that will be used later.First we will derive the mean cycle time E ( C ) . Unlike higher moments of the cycle time,the mean does not depend on where the cycle starts: E ( C ) = E ( S )+ E ( S )1 − ρ . This can easilybe seen, because − ρ is the fraction of time that the server is not working, but switching.The total switch-over time is E ( S ) + E ( S ) .The expected length of a visit to Q i is E ( V i ) = ρ i E ( C ) . The mean length of an intervisitperiod for Q i is E ( I i ) = (1 − ρ i ) E ( C ) . Notice that these expectations do not depend on theservice discipline used. The expected number of type i customers at polling moments doesdepend on the service discipline. For gated service the expected number of type i customersat the beginning of a visit to Q i is λ i E ( C ) . For exhaustive service this is λ i E ( I i ) . The8xpected number of type i customers at the beginning of a visit to Q i +1 is λ i ( E ( V i )+ E ( S i )) for gated service, and λ i E ( S i ) for exhaustive service.Moments of the waiting time distribution for a type i customer at an arbitrary epoch can bederived from the LSTs given by (2.8), (2.9) and (2.10). We only present the first moment:Gated: E ( W i ) = (1 + ρ i ) E ( C i )2 E ( C ) , (2.13)Exhaustive: E ( W i ) = E ( I i )2 E ( I i ) + ρ i − ρ i E ( B i )2 E ( B i ) , = (1 − ρ i ) E ( C ∗ i )2 E ( C ) . (2.14)Notice that the start of C i is the beginning of a visit to Q i , whereas the start of C ∗ i isthe end of a visit. Equations (2.13) and (2.14) are in agreement with Equations (4.1) and(4.2) in [6]. Although at first sight these might seem nice, closed formulas, it should benoted that the expected residual cycle time and the expected residual intervisit time arenot easy to determine, requiring the solution of a large set of equations. MVA is an efficienttechnique to compute mean waiting times, the mean residual cycle time, and also the meanresidual intervisit time. We refer to [26] for an MVA framework for polling models. In this section we study the gated service discipline for a polling system with two queuesand two priority classes in the first queue: high ( H ) and low ( L ) priority customers. Alltype H and L customers that are present at the moment when the server arrives at Q ,will be served during the server’s visit to Q . First all type H customers will be served,then all type L customers. Type H customers arrive at Q according to a Poisson processwith intensity λ H , and have a service requirement B H with LST β H ( · ) . Type L customersarrive at Q with intensity λ L , and have a service requirement B L with LST β L ( · ) . If we donot distinguish between high and low priority customers, we can still use the results fromSection 2 if we regard the system as a polling system with two queues where customers in Q arrive according to a Poisson process with intensity λ := λ H + λ L and have servicerequirement B with LST β ( · ) = λ H λ β H ( · ) + λ L λ β L ( · ) .We follow the same approach as in Section 2. First we study the joint queue lengthdistribution at polling epochs, then the cycle time distribution, followed by the marginalqueue length distribution and waiting time distribution. The last subsection provides thefirst moment of these distributions. 9 .1 Joint queue length distribution at polling epochs Equations (2.2) and (2.3) give the PGFs of the joint queue length distribution at visitbeginnings, V b i ( z , z ) . A type 1 customer entering the system is a type H customer withprobability λ H /λ , and a type L customer with probability λ L /λ . We can express thePGF of the joint queue length distribution in the polling system with priorities, V b i ( · , · , · ) ,in terms of the PGF of the joint queue length distribution in the polling system withoutpriorities, V b i ( · , · ) . Lemma 3.1 V b i ( z H , z L , z ) = V b i (cid:18) λ H z H + λ L z L λ , z (cid:19) . (3.1) Proof:
Let X H be the number of high priority customers present in Q at the beginning of a visitto Q i , i = 1 , . Similarly define X L to be the number of low priority customers present in Q at the beginning of a visit to Q i . Let X = X H + X L . Since the type H / L customersin Q are exactly those H / L customers that arrived since the previous visit beginning at Q i , we know that P ( X H = i, X L = k − i | X = k ) = (cid:18) ki (cid:19) (cid:18) λ H λ (cid:19) i (cid:18) λ L λ (cid:19) k − i . Hence E [ z X H H z X L L | X = k ] = ∞ (cid:88) i =0 ∞ (cid:88) j =0 z iH z jL P ( X H = i, X L = j | X = k )= (cid:18) λ H z H + λ L z L λ (cid:19) k . Finally, V b i ( z H , z L , z ) = ∞ (cid:88) i =0 ∞ (cid:88) j =0 (cid:18) λ H z H + λ L z L λ (cid:19) i z j P ( X = i, X = j )= V b i (cid:18) λ ( λ H z H + λ L z L ) , z (cid:19) . (cid:3) The LST of the cycle time distribution is still given by (2.4) if we define λ := λ H + λ L and β ( · ) := λ H λ β H ( · ) + λ L λ β L ( · ) , because the cycle time does not depend on the order ofservice. 10quation (2.4) is valid for polling systems with queues having any branching type servicediscipline. In the present section we can derive an alternative, shorter expression for γ ( · ) by explicitly using the fact that Q receives gated service. The type 1 (i.e. both H and L )customers present at the visit beginning to Q are those that arrived during the previouscycle: P ( z,
1) = γ ( λ (1 − z )) . By setting ω = λ (1 − z ) , this leads to the followingexpression for the LST of the distribution of C if service in Q is gated: γ ( ω ) = P (1 − ωλ , . (3.2) We first determine the LST of the waiting time distribution for a type L customer, usingthe fact that this customer will not be served until the next cycle (starting at Q ). Thetime from the start of the cycle until the arrival will be called “past cycle time”, denotedby C P . The residual cycle time will be denoted by C R . The waiting time of a type L customer is composed of C R , the service times of all high priority customers that arrivedduring C P + C R , and the service times of all low priority customers that have arrivedduring C P . Let N H ( T ) be the number of high priority customers that have arrived duringtime interval T , and equivalently define N L ( T ) . Theorem 3.2 E (cid:2) e − ωW L (cid:3) = γ ( λ H (1 − β H ( ω )) + λ L (1 − β L ( ω ))) − γ ( ω + λ H (1 − β H ( ω )))( ω − λ L (1 − β L ( ω ))) E ( C ) . Proof: E (cid:2) e − ωW L (cid:3) = E (cid:104) e − ω ( C R + (cid:80) NH ( C P + C R ) i =1 B H,i + (cid:80) NL ( C P ) i =1 B L,i ) (cid:105) = (cid:90) ∞ t =0 (cid:90) ∞ u =0 ∞ (cid:88) m =0 ∞ (cid:88) n =0 E (cid:104) e − ω (cid:80) mi =1 B H,i (cid:105) E (cid:104) e − ω (cid:80) ni =1 B L,i (cid:105) · e − ωu ( λ H ( t + u )) m m ! e − λ H ( t + u ) ( λ L t ) n n ! e − λ L t d P ( C P < t, C R < u )= (cid:90) ∞ t =0 (cid:90) ∞ u =0 e − t ( λ H (1 − β H ( ω ))+ λ L (1 − β L ( ω ))) e − u ( ω + λ H (1 − β H ( ω ))) d P ( C P < t, C R < u )= γ ( λ H (1 − β H ( ω )) + λ L (1 − β L ( ω ))) − γ ( ω + λ H (1 − β H ( ω )))( ω − λ L (1 − β L ( ω ))) E ( C ) . (3.3)For the last step in the derivation of (3.3) we used E [ e − ω P C P − ω R C R ] = E [ e − ω P C ] − E [ e − ω R C ]( ω R − ω P ) E ( C ) , (cid:3) Remark 3.3
The Fuhrmann-Cooper decomposition [13] still holds for the waiting time oftype L customers, because (3.3) can be rewritten into E (cid:2) e − ωW L (cid:3) = (1 − ρ L ) ωω − λ L (1 − β L ( ω )) · γ ( λ H (1 − β H ( ω )) + λ L (1 − β L ( ω ))) − γ ( ω + λ H (1 − β H ( ω )))(1 − ρ L ) ωE ( C ) . (3.4)We recognise the first term on the right-hand side of (3.4) as the LST of the waiting timedistribution of an M/G/ queue with only type L customers. An interpretation of theother two terms on the right-hand side can be found when regarding the polling system asa polling system with three queues ( Q H , Q L , Q ) and no switch-over time between Q H and Q L . The service discipline of this equivalent system is synchronised gated, which is a moregeneral version of gated. The gates for queues Q H and Q L are set simultaneously whenthe server arrives at Q H , but the gate for Q is still set when the server arrives at Q . Inthe following paragraphs we show that the second and third term on the right-hand sideof (3.4) together can be interpreted as E [ (cid:16) − ωλ L (cid:17) N L | I ] , where N L | I is the number of type L customers at a random epoch during the intervisit period of Q L .The expression for the LST of the distribution of the number of type L customers at anarbitrary epoch is determined by first converting the waiting time LST to sojourn timeLST, i.e., multiplying expression (3.4) with β L ( ω ) . Second, we apply the distributionalform of Little’s law [14] to (3.4). This law can be applied because the required conditionsare fulfilled for each customer class (H, L, and 2): the customers enter the system in aPoisson stream, every customer enters the system and leaves the system one at a time inorder of arrival, and for any time t the entry process into the system of customers after time t and the time spent in the system by any customer arriving before time t are independent.The result is: E (cid:2) z N L (cid:3) = (1 − ρ L )(1 − z ) β L ( λ L (1 − z )) β L ( λ L (1 − z )) − z · (cid:101) V c L ( z ) − (cid:101) V b L ( z )(1 − z )( E ( N L | I end ) − E ( N L | I begin )) . (3.5)In this equation (cid:101) V b L ( z ) denotes the PGF of the distribution of the number of type L customers at the beginning of a visit to Q L , and (cid:101) V c L ( z ) denotes the PGF at the completionof a visit to Q L : (cid:101) V b L ( z ) = V b ( β H ( λ L (1 − z )) , z, γ ( λ H (1 − β H ( λ L (1 − z ))) + λ L (1 − z )) , (cid:101) V c L ( z ) = V b ( β H ( λ L (1 − z )) , β L ( λ L (1 − z )) , γ ( λ H (1 − β H ( λ L (1 − z ))) + λ L (1 − β L ( λ L (1 − z )))) . L customers at anarbitrary epoch during the intervisit period of Q L , E [ z N L | I ] . Substitution of ω := λ L (1 − z ) in (3.5), and using ( E ( N L | I end ) − E ( N L | I begin )) = λ L E ( I L ) , shows that the second and thirdterm at the right-hand side of (3.4) together indeed equal E [ (cid:16) − ωλ L (cid:17) N L | I ] .The derivation of the LSTs of W H and W is similar and leads to the following expressions: E (cid:2) e − ωW H (cid:3) = (1 − ρ H ) ωω − λ H (1 − β H ( ω )) · γ ( λ H (1 − β H ( ω ))) − γ ( ω )(1 − ρ H ) ωE ( C ) , (3.6) E (cid:2) e − ωW (cid:3) = (1 − ρ ) ωω − λ (1 − β ( ω )) · γ ( λ (1 − β ( ω ))) − γ ( ω )(1 − ρ ) ωE ( C ) . (3.7) Remark 3.4
Equations (3.6) and (3.7) are equivalent to the LST of W i in a nonprioritypolling system (2.8), which illustrates that the Fuhrmann-Cooper decomposition also holdsfor the waiting time distributions of high priority customers in Q and type 2 customersin a polling system with gated service.Application of the distributional form of Little’s law to these expressions results in: E (cid:2) z N H (cid:3) = (1 − ρ H )(1 − z ) β H ( λ H (1 − z )) β H ( λ H (1 − z )) − z · γ ( λ H (1 − β H ( λ H (1 − z )))) − γ ( λ H (1 − z )) λ H (1 − ρ H )(1 − z ) E ( C ) ,E (cid:2) z N (cid:3) = (1 − ρ )(1 − z ) β ( λ (1 − z )) β ( λ (1 − z )) − z · γ ( λ (1 − β ( λ (1 − z )))) − γ ( λ (1 − z )) λ (1 − ρ )(1 − z ) E ( C ) . Remark 3.5
If the service discipline in Q is not gated, but another branching typeservice discipline that satisfies Property 2.1, (3.7) should be replaced by the more generalexpression (2.7). As mentioned in Section 2.4, we do not focus on moments in this paper, and we onlymention the mean waiting times of type H and L customers. For a type H customer, it isimmediately clear that E ( W H ) = (1 + ρ H ) E ( C , res ) . The mean waiting time for a type L customer can be obtained by differentiating (3.3). This results in: E ( W L ) = (1 + 2 ρ H + ρ L ) E ( C , res ) . These formulas can also be obtained using MVA, as shown in [24].13
Globally gated service
In this section we discuss a polling model with two queues ( Q , Q ) and two priority classes( H and L ) in Q with globally gated service. For this service discipline, only customersthat were present when the server started its visit to Q are served. This feature makes themodel exactly the same as a nonpriority polling model with three queues ( Q H , Q L , Q ) .Although this system does not satisfy Property 2.1, it does satisfy Property 2.2 whichimplies that we can still follow the same approach as in the previous sections. We define the beginning of a visit to Q as the start of a cycle, since this is the moment thatdetermines which customers will be served during the next visits to the queues. Arrivingcustomers will always be served in the next cycle, so the three ( i = H, L, offspring PGFsare: f ( i ) ( z H , z L , z ) = h i ( z H , z L , z )= β i ( λ H (1 − z H ) + λ L (1 − z L ) + λ (1 − z )) , The two ( i = 1 , immigration functions are: g ( i ) ( z H , z L , z ) = σ i ( λ H (1 − z H ) + λ L (1 − z L ) + λ (1 − z )) , Using these definitions, the formula for the PGF of the joint queue length distribution atthe beginning of a cycle is similar to the one found in Section 2: P ( z H , z L , z ) = ∞ (cid:89) n =0 g ( f n ( z H , z L , z )) . (4.1)Notice that in a system with globally gated service it is possible to express the joint queuelength distribution at the beginning of a cycle in terms of the cycle time LST, since allcustomers that are present at the beginning of a cycle are exactly all of the customers thathave arrived during the previous cycle: P ( z H , z L , z ) = γ ( λ H (1 − z H ) + λ L (1 − z L ) + λ (1 − z )) . (4.2) Since only those customers that are present at the start of a cycle, starting at Q , will beserved during this cycle, the LST of the cycle time distribution is γ ( ω ) = σ ( ω ) σ ( ω ) P ( β H ( ω ) , β L ( ω ) , β ( ω )) . (4.3)14ubstitution of (4.2) into this expression gives us the following relation: γ ( ω ) = σ ( ω ) σ ( ω ) · γ ( λ H (1 − β H ( ω )) + λ L (1 − β L ( ω )) + λ (1 − β ( ω ))) . [7] show that this relation leads to the following expression for the cycle time LST: γ ( ω ) = ∞ (cid:89) i =0 σ ( δ ( i ) ( ω )) , where σ ( · ) = σ ( · ) σ ( · ) , and δ ( i ) ( ω ) is recursively defined as follows: δ (0) ( ω ) = ω,δ ( i ) ( ω ) = δ ( δ ( i − ( ω )) , i = 1 , , , . . . ,δ ( ω ) = λ H (1 − β H ( ω )) + λ L (1 − β L ( ω )) + λ (1 − β ( ω )) . For type H and L customers, the expressions for E ( e − ωW H ) and E ( e − ωW L ) are exactly thesame as the ones found in Section 3.3, but with γ ( · ) as defined in (4.3).The expression for E ( e − ωW ) can be obtained with the method used in Section 3.3: E (cid:2) e − ωW (cid:3) = σ ( ω ) · γ ( (cid:80) i = H,L, λ i (1 − β i ( ω ))) − γ ( ω + (cid:80) i = H,L λ i (1 − β i ( ω )))( ω − λ (1 − β ( ω ))) E ( C )= σ ( ω ) · (1 − ρ ) ωω − λ (1 − β ( ω )) · γ ( (cid:80) i = H,L, λ i (1 − β i ( ω ))) − γ ( ω + (cid:80) i = H,L λ i (1 − β i ( ω )))(1 − ρ ) ωE ( C ) . We can use the distributional form of Little’s law to determine the LST of the marginalqueue length distribution of Q : E (cid:2) z N (cid:3) = σ ( λ (1 − z )) (1 − ρ )(1 − z ) β ( λ (1 − z )) β ( λ (1 − z )) − z · γ (cid:16)(cid:80) i = H,L, λ i (1 − β i ( λ (1 − z ))) (cid:17) − γ (cid:16) λ (1 − z ) + (cid:80) i = H,L λ i (1 − β i ( λ (1 − z ))) (cid:17) λ (1 − ρ )(1 − z ) E ( C ) . Remark 4.1
The Fuhrmann-Cooper queue length decomposition also holds for all cus-tomer classes in a polling system with globally gated service.15 .4 Moments
The expressions for E ( W H ) and E ( W L ) from Section 3.4 also hold in a globally gatedpolling system, but with a different mean residual cycle time. We only provide the meanwaiting time of type 2 customers: E ( W ) = E ( S ) + (1 + 2 ρ H + 2 ρ L + ρ ) E ( C , res ) . In this section we study the same polling model as in the previous two sections, but the twoqueues are served exhaustively. The section has the same structure as the other sections,so we start with the derivation of the LST of the joint queue length distribution at pollingepochs, followed by the LST of the cycle time distribution. LSTs of the marginal queuelength distributions and waiting time distributions are provided in the next subsection. Inthe last part of the section the mean waiting time of each customer type is studied.It should be noted that, although we assume that both Q and Q are served exhaustively, amodel in which Q is served according to another branching type service discipline, requiresonly minor adaptations. We can derive the joint queue length distribution at the beginning of a cycle for a pollingsystem with two queues and two priority classes in Q , P ( z H , z L , z ) , directly from (2.1)for P ( z , z ) . Similar to the proof of Lemma 3.1, we can prove that P ( z H , z L , z ) = P (cid:18) λ ( λ H z H + λ L z L ) , z (cid:19) . The same holds for V b ( · , · , · ) and visit completion epochs V c i ( · , · , · ) , for i = 1 , . For the cycle time starting with a visit to Q , (2.4) is still valid. However, when studyingthe waiting time of a specific customer type in an exhaustively served queue, it is convenientto consider the completion of a visit to Q as the start of a cycle. Hence, in this sectionthe notation C ∗ , or the LST of its distribution, γ ∗ ( · ) , refers to the cycle time starting atthe completion of a visit to Q . Equation (2.5) gives the LST of the distribution of C ∗ .Using the fact that customers in Q are served exhaustively, we can find an alternative,compact expression for γ ∗ ( · ) . The type 1 (i.e. both type H and L customers) customers at16he beginning of a visit to Q are exactly those type 1 customers that have arrived duringthe previous intervisit time: P ( z,
1) = (cid:101) I ( λ (1 − z )) . Hence, by setting ω = λ (1 − z ) , weget (cid:101) I ( ω ) = P (1 − ωλ , , and thus by (2.11), γ ∗ ( ω ) = P ( π ( ω ) − ωλ , . (5.1) Analysis of the model with exhaustive service requires a different approach. The keyobservation, made by [13], is that a nonpriority polling system from the viewpoint of atype i customer is an M/G/ queue with multiple server vacations. This implies thatthe Fuhrmann-Cooper decomposition can be used, even though the intervisit times arestrongly dependent on the visit times. The M/G/ queue with priorities and vacations canbe analysed by modelling the system as a special version of the nonpriority M/G/ queuewith multiple server vacations, and then applying the results from Fuhrmann and Cooper.This approach has been used by [15] who used the concept of delay cycles , and also by [21]who used level crossing analysis ; see also [23]. We apply Kella and Yechiali’s approachto the polling model under consideration to find the waiting time LST for type H and L customers. In [15] systems with single and multiple vacations, preemptive resume andnonpreemptive service are considered. In the present paper we do not consider preemptiveresume, so we only use results from the case labelled as NPMV (nonpreemptive, multiplevacations) in [15]. We consider the system from the viewpoint of a type H and type L customer separately to derive E [ e − ωW H ] and E [ e − ωW L ] .From the viewpoint of a type H customer and as far as waiting times are concerned, apolling system is a nonpriority single server system with multiple vacations. The vacationcan either be the intervisit period I , or the service of a type L customer. The LSTs ofthese two types of vacations are: E [ e − ωI ] = P (1 − ω/λ , , (5.2) E [ e − ωB L ] = β L ( ω ) . Equation (5.2) follows immediately from the fact that the number of type 1 (i.e. both Hand L) customers at the beginning of a visit to Q is the number of type 1 customers thathave arrived during the previous intervisit period: P ( z,
1) = E [ e − ( λ (1 − z )) I ] .We now use the concept of delay cycles, introduced in [15], to find the waiting time LSTof a type H customer. The key observation is that an arrival of a tagged type H customerwill always take place within either an I H cycle, or an L H cycle. An I H cycle is a cyclethat starts with an intervisit period for Q , followed by the service of all type H customersthat have arrived during the intervisit period, and ends at the moment that no type H customers are left in the system. Notice that at the start of the intervisit period, no type H customers were present in the system either. An L H cycle is a similar cycle, but starts17ith the service of a type L customer. This cycle also ends at the moment that no type H customers are left in the system.The fraction of time that the system is in an L H cycle is ρ L − ρ H , because type L customersarrive with intensity λ L . Each of these customers will start an L H cycle and the length ofan L H cycle equals E ( B L )1 − ρ H : E ( L H cycle ) = E ( B L ) + λ H E ( B L ) E ( BP H )= E ( B L ) + λ H E ( B L ) E ( B H )1 − ρ H = (1 + ρ H − ρ H ) E ( B L ) = E ( B L )1 − ρ H , where E ( BP H ) is the mean length of a busy period of type H customers.The fraction of time that the system is in an I H cycle, is − ρ L − ρ H = − ρ − ρ H . This resultcan also be obtained by using the argument that the fraction of time that the system is inan intervisit period is the fraction of time that the server is not serving Q , which is equalto − ρ . A cycle which starts with such an intervisit period and stops when all type H customers that arrived during the intervisit period and their type H descendants have beenserved, has mean length E ( I )+ λ H E ( I ) E ( BP H ) = E ( I )1 − ρ H . This also leads to the conclusionthat − ρ − ρ H is the fraction of time that the system is in an I H cycle. A customer arrivingduring an I H cycle views the system as a nonpriority M/G/ queue with multiple servervacations I ; a customer arriving during an L H cycle views the system as a nonpriority M/G/ queue with multiple server vacations B L .[13] showed that the waiting time of a customer in an M/G/ queue with server vacationsis the sum of two independent quantities: the waiting time of a customer in a corresponding M/G/ queue without vacations, and the residual vacation time. Hence, the LST of thewaiting time distribution of a type H customer is: E [ e − ωW H ] = (1 − ρ H ) ωω − λ H (1 − β H ( ω )) · (cid:34) − ρ − ρ H · − (cid:101) I ( ω ) ωE ( I ) + ρ L − ρ H · − β L ( ω ) ωE ( B L ) (cid:35) . (5.3)Equation (5.3) is in accordance with the more general equation in Section 4.1 in [15]. Remark 5.1
The LST of the distribution of the waiting time of a high priority customerin a two priority
M/G/ queue without vacations is E [ e − ωW H | M/G/ ] = (1 − ρ ) ω + λ L (1 − β L ( ω )) ω − λ H (1 − β H ( ω )) , (5.4)see, e.g., Equation (3.85) in [10], Chapter III.3. Equation (5.4) can be rewritten to (5.3),with − (cid:101) I ( ω ) ωE ( I ) replaced by 1. Hence, the waiting time distribution of a high priority customerin a two priority M/G/ queue equals the waiting time distribution of a customer in anonpriority M/G/ queue with only type H customers, where the server goes on a vacation B L with probability ρ L − ρ H . 18 emark 5.2 Substitution of (2.12) in (5.3) expresses E [ e − ωW H ] in terms of the LST ofthe cycle time distribution starting at a visit completion to Q , γ ∗ ( · ) : E [ e − ωW H ] = 1 − γ ∗ ( ω − λ H (1 − β H ( ω )) − λ L (1 − β L ( ω ))) + λ L (1 − β L ( ω )) E ( C )( ω − λ H (1 − β H ( ω ))) E ( C ) . (5.5)The concept of cycles is not really needed to model the system from the perspective of atype L customer, because for a type L customer the system merely consists of I HL cycles.An I HL cycle is the same as an I H cycle, discussed in the previous paragraphs, exceptthat it ends when no type H or L customers are left in the system. So the system canbe modelled as a nonpriority M/G/ queue with server vacations. The vacation is theintervisit time I , plus the service times of all type H customers that have arrived duringthat intervisit time and their type H descendants. We will denote this extended intervisittime by I ∗ with LST (cid:101) I ∗ ( ω ) = (cid:101) I ( ω + λ H (1 − π H ( ω ))) . The mean length of I ∗ equals E ( I ∗ ) = E ( I )1 − ρ H .We also have to take into account that a busy period of type L customers might beinterrupted by the arrival of type H customers. Therefore the alternative system thatwe are considering will not contain regular type L customers, but customers still arrivingwith arrival rate λ L , whose service time equals the service time of a type L customer inthe original model, plus the service times of all type H customers that arrive during thisservice time, and all of their type H descendants. The LST of the distribution of thisextended service time B ∗ L is β ∗ L ( ω ) = β L ( ω + λ H (1 − π H ( ω ))) . This extended service time is often called completion time in the literature. In this alterna-tive system, the mean service time of these customers equals E ( B ∗ L ) = E ( B L )1 − ρ H . The fractionof time that the system is serving these customers is ρ ∗ L = ρ L − ρ H = 1 − − ρ − ρ H .Now we use the results from the M/G/ queue with server vacations (starting with theFuhrmann-Cooper decomposition) to determine the LST of the waiting time distributionfor type L customers: E [ e − ωW L ] = (1 − ρ ∗ L ) ωω − λ L (1 − β ∗ L ( ω )) · − (cid:101) I ∗ ( ω ) ωE ( I ∗ )= (1 − ρ )( ω + λ H (1 − π H ( ω ))) ω − λ L (1 − β L ( ω + λ H (1 − π H ( ω )))) · − (cid:101) I ( ω + λ H (1 − π H ( ω )))( ω + λ H (1 − π H ( ω ))) E ( I ) . (5.6)The last term of (5.6) is the LST of the distribution of the residual intervisit time, plus thetime that it takes to serve all type H customers and their type H descendants that arriveduring this residual intervisit time. The first term of (5.6) is the LST of the waiting timedistribution of a low-priority customer in an M/G/ queue with two priorities, withoutvacations (see e.g. (3.76) in [10], Chapter III.3).19 emark 5.3 The
M/G/ queue with two priorities can be viewed as a nonpriority M/G/ queue with vacations, if we consider the waiting time of type L customers. We only needto rewrite the first term of (5.6): E [ e − ωW L | M/G/ ] = (1 − ρ )( ω + λ H (1 − π H ( ω ))) ω − λ L (1 − β L ( ω + λ H (1 − π H ( ω ))))= (1 − ρ ∗ L ) ωω − λ L (1 − β ∗ L ( ω )) · − ρ − ρ ∗ L · ω + λ H (1 − π H ( ω )) ω = E [ e − ωW ∗ L | M/G/ ] · (cid:20) (1 − ρ H ) + ρ H − π H ( ω ) ωE ( BP H ) (cid:21) , where E [ e − ωW ∗ L | M/G/ ] is the LST of the waiting time distribution of a customer in an M/G/ queue where customers arrive at intensity λ L and have service requirement LST β L ( ω + λ H (1 − π H ( ω ))) . So with probability − ρ H the waiting time of a customer is thewaiting time in an M/G/ queue with no vacations, and with probability ρ H the waitingtime of a customer is the sum of the waiting time in an M/G/ queue and the residuallength of a vacation, which is a busy period of type H customers. Remark 5.4
Substitution of (2.12) in (5.6) leads to a different expression for E [ e − ωW L ] : E [ e − ωW L ] = 1 − γ ∗ ( ω − λ L (1 − β L ( ω + λ H (1 − π H ( ω )))))( ω − λ L (1 − β L ( ω + λ H (1 − π H ( ω ))))) E ( C )= E [ e − ( ω − λ L (1 − β L ( ω + λ H (1 − π H ( ω ))))) C ∗ , res ] . (5.7)The waiting time of type 2 customers is not affected at all by the fact that Q containsmultiple classes of customers, so (2.9) is still valid for E ( e − ωW ) .We will refrain from mentioning the PGFs of the marginal queue length distributions here,because they can be obtained by applying the distributional form of Little’s law as we havedone before. The mean waiting times for high and low priority customers can be found by differentiationof (5.3) and (5.6): E ( W H ) = ρ H E ( B H, res ) + ρ L E ( B L, res )1 − ρ H + 1 − ρ − ρ H E ( I , res ) ,E ( W L ) = ρ H E ( B H, res ) + ρ L E ( B L, res )(1 − ρ H )(1 − ρ ) + 11 − ρ H E ( I , res ) . E ( W H ) = (1 − ρ ) − ρ H E ( C ∗ )2 E ( C ) ,E ( W L ) = (1 − ρ ) (1 − ρ H )(1 − ρ ) E ( C ∗ )2 E ( C )= (cid:18) − ρ L − ρ H (cid:19) E ( C ∗ )2 E ( C ) . Consider a polling system with two queues, and assume exponential service times andswitch-over times. Suppose that λ = , λ = , E ( B ) = E ( B ) = 1 , E ( S ) = E ( S ) = 1 .The workload of this polling system is ρ = . This example is extensively discussed in[26] where MVA was used to compute mean waiting times and mean residual cycle timesfor the gated and exhaustive service disciplines.In this example we show that the performance of this system can be improved by givinghigher priority to jobs with smaller service times. We define a threshold t and divide thejobs into two classes: jobs with a service time less than t receive high priority, the otherjobs receive low priority. In Figures 1 and 2 the mean waiting times of customers in Q are shown as a function of the threshold t . The following four cases are distinguished: • the mean waiting time of the low priority customers in Q (indicated as “Type L”); • the mean waiting time of the high priority customers in Q (indicated as “Type H”); • a weighted average of the above two mean waiting times: λ L λ E ( W L ) + λ H λ E ( W H ) (indicated as “Type 1 with priorities”). This can be interpreted as the mean waitingtime of an arbitrary customer in Q ; • the mean waiting time of an arbitrary customer in Q if no priority rules would beapplied to this queue (indicated as “Type 1 no priorities”). In this situation there isno such thing as high and low priority customers, so the mean waiting time does notdepend on t , and has already been computed in [26].The figures show that a unique optimal threshold exists that minimises the mean weightedwaiting time for customers in Q . This value depends on the service discipline used andis discussed in [24]. In this example the optimal threshold is 1 for gated, and 1.38 forexhaustive. Figure 1 confirms that the mean waiting times for type H and L customersin the gated model only differ by a constant value: E ( W L ) − E ( W H ) = ρ E ( C , res ) . Forglobally gated service no figure is included, because we again have E ( W L ) − E ( W H ) = E ( C , res ) . The mean residual cycle time is different from the one in the gated model, butthis does not affect the optimal threshold which is still t = 1 .In the exhaustive model we have the following relation: E ( W L ) − E ( W H ) = ρ (1 − ρ )1 − ρ H E ( C ∗ , res ) . If we increase threshold t , the fraction of customers in Q that receive high priority grows,and so does their mean service time. This means that ρ H increases as t increases, so E ( W L ) − E ( W H ) gets bigger, which can be seen in Figure 2. Notice that E ( W H ) E ( W L ) = 1 − ρ ,so it does not depend on t . t E H W L Type HType 1 with prioritiesType 1 no prioritiesType L
Figure 1: Mean waiting time of customers in Q in the gated polling system, versus thresh-old t .It is interesting to also consider the variance, or rather the standard deviation of the waitingtime. Figures 3 and 4 show the standard deviation of the type H and L customers versusthe threshold t . The figures also show the standard deviation of an arbitrary customer in Q , with and without priorities. The figures indicate that the waiting times in the gatedsystem have smaller standard deviations than in the exhaustive case. In this example, theintroduction of priorities affects the standard deviation of an arbitrary type 1 customeronly slightly. However, it is interesting to zoom in to investigate the influence of threshold t . Figure 5 contains zoomed versions of Figures 3 and 4 and indicates that the threshold t that minimises the overall mean waiting time of type customers in the priority systemdoes not minimise the standard deviation. In fact, changing threshold t affects the entireservice time distributions B H and B L , which results in two local minima for the standarddeviation as function of threshold t . 22 t E H W L Type HType 1 with prioritiesType 1 no prioritiesType L
Figure 2: Mean waiting time of customers in Q in the exhaustive polling system, versusthreshold t . t H W L Type HType 1 with prioritiesType 1 no prioritiesType L
Figure 3: Standard deviation of the waiting time of customers in Q in the gated pollingsystem, versus threshold t . The polling system studied in the present paper leaves many possibilities for extensions orvariations. In this section we discuss some of them.
Multiple queues and priority levels.
Probably the most obvious extension of themodel under consideration, is a polling system with any number of queues and any numberof priority levels in each queue. In recent research [2], we have discovered that such a polling23 t H W L Type HType 1 with prioritiesType 1 no prioritiesType L
Figure 4: Standard deviation of the waiting time of customers in Q in the exhaustivepolling system, versus threshold t . t H W L t H W L Figure 5: Zoomed versions of Figures 3 (left) and 4 (right).model can be analysed in detail. Each queue can have its own service discipline, eitherexhaustive or (synchronised) gated.
Preemptive resume.
In the present paper, the service of low priority customers is notinterrupted by the arrival of a high priority customer. If we allow for service interruptions,these would only take place in a queue with exhaustive service, since (globally) gatedservice forces high priority customers to wait behind the gate. We note that allowing serviceinterruptions does not affect the joint queue length distributions at polling instants, nor thecycle time. Also the waiting time of low priority customers is unaffected (but they mighthave a longer sojourn time ). It only affects the waiting time of high priority customers,because they do not have to wait for a residual service time of a low priority customer. TheLST of the waiting time distribution of a high priority customer if service is preemptiveresume, is: E [ e − ωW H ] = (1 − ρ H ) ωω − λ H (1 − β H ( ω )) · (cid:34) − ρ − ρ H · − (cid:101) I ( ω ) ωE ( I ) + ρ L − ρ H (cid:35) . ixed gated/exhaustive service. In the present paper, customers in Q receive ei-ther exhaustive or (globally) gated service. One may consider serving each priority levelaccording to a different service discipline. In [1], high priority customers receive exhaus-tive service, whereas low priority customers receive gated service. This gives high prioritycustomers an additional advantage, but it turns out that for low priority customers thisstrategy may be better than, e.g., gated service for all priority levels. A mixture of globallygated service for low priority customers and exhaustive service for high priority customerscan be analysed similarly.The “opposite” strategy, where low priority customers are served exhaustively and highpriority customers are served according to the gated service discipline is easier to analyse,since we can model it as a nonpriority polling model with Q replaced by two queues, Q H and Q L , containing the type H and type L customers and having gated and exhaustiveservice respectively. Partially gated.
A variant of the gated service discipline is partially gated service: everycustomer, type H or L , standing in front of the gate is served during a visit with a fixedprobability p , and is not served with probability − p . The probability p might evendepend on the customer type. Whether a rejected customer is eligible for service in thenext cycle, or leaves the system, does not matter. Both situations can be analysed. Different polling sequences.
We assume that the server alternates between Q and Q .A different way of introducing priorities to a polling system is by increasing the frequencyof visits to a queue within a cycle. One can, e.g., decide to visit Q two consecutive timesif gated service is used. Or one can think of a system where the server switches to Q j aftercompleting a visit to Q i with probability p ij . Large setup times. [25] establishes fluid limits for polling systems with any branchingtype service discipline and deterministic switch-over times tending to infinity. The scaledwaiting time distribution is shown to converge to a uniform distribution with bounds thatcan be computed explicitly. The results are relevant to applications in production systems,where large setup times are common. These fluid limits can also be computed for thepolling model that is discussed in the present paper and give explicit insight in when eachof the discussed service disciplines is optimal.
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