Closed-Form Waiting Time Approximations for Polling Systems
CClosed-Form Waiting Time Approximations for Polling Systems ∗ M.A.A. Boon † [email protected] E.M.M. Winands ‡ [email protected] I.J.B.F. Adan † [email protected]. van Wijk § [email protected], 2010 Abstract
A typical polling system consists of a number of queues, attended by a single serverin a fixed order. The vast majority of papers on polling systems focusses on Poissonarrivals, whereas very few results are available for general arrivals. The current studyis the first one presenting simple closed-form approximations for the mean waiting timesin polling systems with renewal arrival processes, performing well for all workloads. Theapproximations are constructed using heavy traffic limits and newly developed light trafficlimits. The closed-form approximations may prove to be extremely useful for system de-sign and optimisation in application areas as diverse as telecommunication, maintenance,manufacturing and transportation.
Keywords:
Polling, waiting times, queue lengths, approximation
Polling systems are queueing systems consisting of multiple queues, visited by a single server- typically in a fixed, cyclic order. They find their origin in many real-life applications, e.g.(computer) communication, production and manufacturing environments, traffic and trans-portation. For a good literature overview of polling systems and their applications, we referto surveys of, e.g., Takagi [21], Levy and Sidi [14], and Vishnevskii and Semenova [24]. Whenstudying literature on polling systems, it rapidly becomes apparent that the computationof the distributions and moments of the waiting times and marginal queue lengths is verycumbersome. Closed form expressions do not exist, and even when one specifies the numberof queues and solves the set of equations that leads to the mean waiting times, the obtained ∗ The research was done in the framework of the BSIK/BRICKS project, and of the European Network ofExcellence Euro-NF. † Eurandom and Department of Mathematics and Computer Science, Eindhoven University of Technology,P.O. Box 513, 5600MB Eindhoven, The Netherlands ‡ Department of Mathematics, Section Stochastics, VU University, De Boelelaan 1081a, 1081HV Amster-dam, The Netherlands § Department of Industrial Engineering & Innovation Sciences and Department of Mathematics and Com-puter 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 xpressions are still too lengthy and complicated to interpret directly. Numerical procedures,both approximate and exact, have been developed in the past to compute these performancemeasures. However, these methods have several drawbacks. Firstly, they are not transpar-ent and act as a kind of black box. It is, for instance, rather difficult to study the impactof parameters like the occupation rate and the service level. Secondly, these procedures arecomputationally complex and hard, if not impossible, to implement in a standard spreadsheetprogram commonly used on the work floor. Finally, the vast majority of standard methodsfocusses on Poisson arrival processes, which may not be very realistic in many applicationareas. In the present paper we study polling systems in which the arrival streams are not(necessarily) Poisson, i.e., the interarrival times follow a general distribution. The goal isto derive closed-form approximate solutions for the mean waiting times and mean marginalqueue lengths, which can be computed by simple spreadsheet calculations.Our approach in developing an approximation for the mean waiting times uses novel devel-opments in polling literature. Recently, a heavy traffic (HT) limit has been developed forthe mean waiting times as the system becomes saturated [6, 17, 23]. In the present paperwe derive an approximation for the light traffic (LT) limit, i.e. as the load decreases to zero,which is exact for Poisson arrivals. The main idea is to create an interpolation between theLT limit and the HT limit. This interpolation yields good results, and has several nice prop-erties, like satisfying the Pseudo Conservation Law (PCL), and being exact for symmetricsystems with Poisson arrivals and in many limiting cases. These properties are described inmore detail in the present paper. In polling literature, several alternative approximationshave been developed before, most of which assume Poisson arrivals. For polling systems withPoisson arrivals and gated or exhaustive service, the best results, by far, are obtained by anapproximation based on the PCL (see, e.g., [3, 7, 11]). Fischer et al. [8] study an approxima-tion for the mean waiting times in polling systems, which is also based on an interpolationbetween (approximate) LT and HT limits. Their approach, however, is applied to a systemwith Poisson arrivals and time-limited service. Hardly any closed-form approximations existfor non-Poisson arrivals. The few that exist, perform well in specific limiting cases, e.g., underHT conditions [17, 23], or if switch-over times become very large [27, 28], but performancedeteriorates rapidly if these limiting conditions are abandoned, in contrast to the approxi-mation developed in the present paper. We show in an extensive numerical study that thequality of our approximation can be compared to the PCL approximation for systems withPoisson arrivals, but provides good results as well for systems with renewal arrivals.Because of its simple form, the approximation function is very suitable for optimisation pur-poses and implementation in a spreadsheet. Although only the mean waiting times of systemswith exhaustive or gated service are studied, the results can be extended to higher momentsand general branching-type service disciplines. Polling systems with polling tables and/orbatch service can also be analysed in a similar manner.The structure of the present paper is as follows: the next section introduces the model andthe required notation, and states the main result. Section 3 illustrates how this main result isobtained, while Section 4 provides results on the accuracy of the approximation for a large setof combinations of input parameter values. The last section discusses further research topicsand possible extensions of the model. 2 Model description and main result
The model under consideration is a polling system consisting of N queues, Q , . . . , Q N , withrenewal arrival processes. Indices throughout the present paper are understood to be modulo N : Q N +1 actually refers to Q . Whenever a server switches from Q i to Q i +1 , a randomswitch-over time S i is incurred. The generic service requirement of a customer arriving in Q i ,also referred to as a type i customer, is denoted by the random variable B i . We make theusual independence assumptions for polling systems; the interarrival times, service times andswitch-over times are all independent. The moment at which the server switches from onequeue to the next queue, is determined by the service discipline of the queue that is beingserved. In the present paper we focus on polling systems in which each queue is either servedaccording to the gated service discipline, which states that during the course of a visit of theserver to Q i , only those type i customers are served that were present at the beginning ofthat visit, or according to the exhaustive service discipline, which means that the server keepson serving type i customers until Q i is empty, before switching to Q i +1 .We regard several variables as a function of the load ρ in the system. Scaling is done bykeeping the service time distributions fixed, and varying the interarrival times. For eachvariable x that is a function of the load in the system, ρ , its value evaluated at ρ = 1 isdenoted by ˆ x . For ρ = 1, the generic interarrival time of the stream in Q i is denoted byˆ A i . Reducing the load ρ is done by scaling the interarrival times, i.e., taking the randomvariable A i := ˆ A i /ρ as generic interarrival time at Q i . After scaling, the load at Q i becomes ρ i = ρ E [ B i ] E [ ˆ A i ] . The (scaled) rate of the arrival stream at Q i is defined as λ i = 1 / E [ A i ]. Similarly,we define arrival rates ˆ λ i = 1 / E [ ˆ A i ], and proportional load at Q i , ˆ ρ i = ρ i ρ (“proportional”because (cid:80) Ni =1 ˆ ρ i = 1). The system is assumed to be stable, so ρ is varied between 0 and 1.We use B to denote the generic service requirement of an arbitrary customer entering thesystem, with E [ B k ] = (cid:80) Ni =1 ˆ λ i E [ B ki ] (cid:80) Nj =1 ˆ λ j for any integer k >
0, and S = (cid:80) Ni =1 S i denotes the totalswitch-over time in a cycle. Finally, the (equilibrium) residual length of a random variable X is denoted by X res , with E [ X res ] = E [ X ] / E [ X ].We now present the main result of this paper, which is a closed-form approximation formulafor the mean waiting time E [ W i ] of a type i customer as a function of ρ : E [ W i, app ] = K ,i + K ,i ρ + K ,i ρ − ρ , i = 1 , . . . , N. (2.1)The constants K ,i , K ,i , and K ,i depend on the input parameters and the service discipline.If all queues receive exhaustive service, the constants become: K ,i = E [ S res ] , (2.2) K ,i = ˆ ρ i (cid:0) E [ ˆ A i ]ˆ g i (0) − (cid:1) E [ B res i ] + E [ B res ] + ˆ ρ i (cid:0) E [ S res ] − E [ S ] (cid:1) − E [ S ] N − (cid:88) j =0 j (cid:88) k =0 ˆ ρ i + k V ar[ S i + j ] , (2.3) K ,i = 1 − ˆ ρ i (cid:80) Nj =1 ˆ λ j (cid:16) V ar[ B j ] + ˆ ρ j V ar[ ˆ A j ] (cid:17)(cid:80) Nj =1 ˆ ρ j (1 − ˆ ρ j ) + E [ S ] − K ,i − K ,i . (2.4)3f all queues receive gated service, we get: K ,i = E [ S res ] , (2.5) K ,i = ˆ ρ i (cid:0) E [ ˆ A i ]ˆ g i (0) − (cid:1) E [ B res i ] + E [ B res ] + ˆ ρ i E [ S res ] − E [ S ] N − (cid:88) j =0 j (cid:88) k =0 ˆ ρ i + k V ar[ S i + j ] , (2.6) K ,i = 1 + ˆ ρ i (cid:80) Nj =1 ˆ λ j (cid:16) V ar[ B j ] + ˆ ρ j V ar[ ˆ A j ] (cid:17)(cid:80) Nj =1 ˆ ρ j (1 + ˆ ρ j ) + E [ S ] − K ,i − K ,i . (2.7)The term ˆ g i ( t ) is the density of ˆ A i , the interarrival times at ρ = 1. This term is discussed inmore detail in the next section, but for practical purposes it is useful to know that E [ ˆ A i ]ˆ g i (0)can be very well approximated by E [ ˆ A i ]ˆ g i (0) ≈ cv Ai cv Ai +1 if cv A i > , (cid:0) cv A i (cid:1) if cv A i ≤ , where cv A i is the squared coefficient of variation (SCV) of A i (and, hence, also of ˆ A i ). Notethat this simplification results in an approximation that requires only the first two momentsof each input variable (i.e., service times, switch-over times, and interarrival times). Remark 2.1
In case of Poisson arrivals, the constants K ,i and K ,i simplify considerably.E.g., for exhaustive service they simplify to: K Poisson ,i = E [ B res ] + ˆ ρ i (cid:0) E [ S res ] − E [ S ] (cid:1) − E [ S ] N − (cid:88) j =0 j (cid:88) k =0 ˆ ρ i + k V ar[ S i + j ] ,K Poisson ,i =(1 − ˆ ρ i ) (cid:32) E [ B res ] (cid:80) Nj =1 ˆ ρ j (1 − ˆ ρ j ) + E [ S ]2 (cid:33) − K ,i − K Poisson ,i . The derivation of this approximative formula for the mean waiting time is the topic of the nextsection. An approximation for the mean queue length at Q i , E [ L i ] is obtained by applicationof Little’s Law to the sojourn time of type i customers, i.e. the waiting time plus the servicetime. As a function of ρ , we have E [ L i, app ] = ρ E [ W i, app ] + E [ B i ] E [ ˆ A i ] . Approximation (2.1) is an interpolation approximation based on LT and HT limits. In thenext subsection we first provide a motivation for this approach.4 .1 Generic interpolation function
In its generic form, the interpolation approximation proceeds as follows; see [18, 20]. Consideran open queueing system with load ρ . Let f ( ρ ), 0 ≤ ρ <
1, be some function of the queueingsystem (such as the mean waiting time), which is assumed to be analytic on [0 , f ( ρ ) can be expressed as f ( ρ ) = ∞ (cid:88) n =0 f ( n ) (0) n ! ρ n , ≤ ρ < , where f ( n ) ( ρ ) denotes the n th derivative f ( ρ ). Usually, f ( ρ ) is intractable, but it may bepossible to derive partial information about f ( ρ ), such as the light traffic limits f ( n ) (0) for n =0 , , . . . , k and the “canonical” heavy traffic limit h = lim ρ → (1 − ρ ) f ( ρ ). For examples, see[9, 18, 25], where based on the partial information, an approximation for f ( ρ ) is constructedof the form ˜ f ( ρ ) = q ( ρ )1 − ρ , (3.1)where q ( ρ ) is the ( k + 1)st degree polynomial, uniquely determined by the requirement that˜ f ( ρ ) has to match everything that is known about f ( ρ ), i.e., ˜ f ( n ) (0) = f ( n ) (0) for n =0 , , . . . , k and lim ρ → (1 − ρ ) ˜ f ( ρ ) = h . The heavy traffic limit implies (see Prop. 1 in [20]),lim n →∞ f ( n ) (0) n ! = h. This suggests that, in the Taylor series, f ( n ) (0) for n > k can be approximated by n ! h . Thus,combined with knowledge of the light traffic limits f ( n ) (0) for n = 0 , , . . . , k , the followingnew approximation can be produced,¯ f ( ρ ) = k (cid:88) n =0 f ( n ) (0) n ! ρ n + h ρ k +1 − ρ . Interestingly, this seemingly different approximation ¯ f ( ρ ) is identical to the interpolationapproximation ˜ f ( ρ ), confirming the notion that they are the “natural” approximation for f ( ρ ), given the partial information. In [18] the interpolation approximation (3.1) is shownto work extremely well for several examples. The present paper applies approximation (3.1)to the new setting of polling systems with general renewal arrivals, for which no analyticexpressions are known for the mean waiting times. Choosing f ( ρ ) as the mean waiting timeof a type i customer, we derive new (approximations for the) light traffic limits f (0) and thefirst derivative f (cid:48) (0), which together with the heavy traffic limit, yield an interpolation witha quadratic polynomial q ( ρ ); see (2.1).In Sections 3.2 and 3.3 we derive the LT and HT limits, respectively. The interpolationapproximation, matching these limits, is presented in Section 3.4. Finally, in Section 3.5, weshow that the interpolation approximation also matches known exact results for mean waitingtimes in polling systems. In particular, we prove that the form (3.1) is crucial for satisfyingthe pseudo-conservation law for all loads ρ . An extensive numerical validation is the topicof Section 4, showing that the interpolation approximation works extremely well for pollingsystems. 5 .2 Light traffic The mean waiting times in the polling model under consideration in light-traffic, have beenstudied in Blanc and Van der Mei [2], under the assumption of Poisson arrivals. They obtainexpressions for the mean waiting times in light traffic that are exact up to (and including) first-order terms in ρ . These expressions have been found by carefully inspecting numerical resultsobtained with the Power-Series Algorithm, but no proof is provided. In the present sectionwe shall not only prove the correctness of the light-traffic results in a system with Poissonarrivals, but also use them as base for an approximation for the mean waiting times in pollingsystems with renewal interarrival times. The key ingredient to the LT analysis of a pollingsystem, is the well-known Fuhrmann-Cooper decomposition [10]. It states that in a vacationsystem with Poisson arrivals the queue length of a customer is the sum of two independentrandom variables: the number of customers in an isolated M/G/ V i to denotethe length of a visit period to Q i , and I i to denote the length of the intervisit period, i.e. thetime that the server is away between two successive visits to Q i . Using C i to denote the cycletime, starting at a visit beginning to Q i , we have E [ V i ] = ρ i E [ C i ] and E [ I i ] = (1 − ρ i ) E [ C i ]. Itis well-known that the mean cycle time in polling systems, unlike higher moments, does notdepend on the starting point: E [ C i ] = E [ C ] = E [ S ]1 − ρ .The Fuhrmann-Cooper decomposition, applied to the mean waiting time, results in:exhaustive: E [ W i ] = E [ W i,M/G/ ] + E [ I res i ] , (3.2)gated: E [ W i ] = E [ W i,M/G/ ] + E [ I res i ] + E [ V i I i ] E [ I i ] . (3.3)For our approximation, we assume that this decomposition also holds for renewal arrivalprocesses in light traffic. Determining the LT limit of the mean waiting time, E [ W LT i ], in apolling system with exhaustive or gated service is based on the following two-step approach.The first step is to find the LT limit of E [ W i,GI/G/ ], the mean waiting time of a GI/G/ i customers in isolation, i = 1 , . . . , N . The second step is determining E [ I res i ], the mean residual intervisit time of Q i , and E [ V i I i ] E [ I i ] , the mean visit time of Q i giventhat it is being observed at a random epoch during the following intervisit time. Remark 3.1
Bertsimas and Mourtzinou [1] state that the decompositions (3.2) and (3.3)also hold for polling systems with Mixed Generalised Erlang arrivals. However, simulationand exact analysis of some simple cases indicate that the decomposition result is not valid forthe mean waiting times.For the LT limit of the mean waiting time in a
GI/G/ ρ i ↓ E [ W i,GI/G/ ] ρ i = 1 + cv B i E [ ˆ A i ]ˆ g i (0) E [ B i ] , (3.4)where cv B i is the SCV of the service times, and ˆ g i ( t ) is the density of the interarrival times ˆ A i .For practical purposes, it may be more convenient to express ˆ g i (0) in terms of the density of6 i , the generic interarrival time of Q i in the scaled situation.The relation between the densityof the scaled interarrival times A i (= ˆ A i /ρ ), denoted by g i ( t ), and the density of ˆ A i , ˆ g i ( t ), issimply: g i ( t ) = ρ ˆ g i ( ρt ). This means that the term E [ ˆ A i ]ˆ g i (0) can be rewritten as E [ ˆ A i ]ˆ g i (0) = E [ A i ] g i (0) . Because of this equality, in the remainder of the paper we might use either notation. Sincedetermining E [ ˆ A i ]ˆ g i (0) is a required step in the computation of our approximation for E [ W i ],we give some practical examples. Example 1
If the scaled interarrival times A i are exponentially distributed with parameter λ i := 1 / E [ A i ], we have g i ( t ) = λ i e − λ i t . This implies that E [ A i ] g i (0) = 1. Example 2
In this example we assume that A i follows a H distribution with balancedmeans. The SCV of A i is denoted by cv A i . The density of this hyper-exponential distributionis (see, e.g., [22]) g i ( t ) = pµ e − µ t + (1 − p ) µ e − µ t , with p = 12 (cid:32) (cid:115) cv A i − cv A i + 1 (cid:33) ,µ = 1 E [ A i ] (cid:32) (cid:115) cv A i − cv A i + 1 (cid:33) ,µ = 1 E [ A i ] (cid:32) − (cid:115) cv A i − cv A i + 1 (cid:33) . This leads to E [ A i ] g i (0) = 1 + cv A − cv A +1 = 2 cv A cv A +1 . Example 3
Now we assume that the interarrival times follow a mixed Erlang distribution.The density of the scaled interarrival times is: g i ( t ) = p µ k − t k − ( k − − µt + (1 − p ) µ k t k − ( k − − µt , i.e., a mixture of an Erlang( k −
1) and an Erlang( k ) distribution with k = (cid:38) cv A i (cid:39) ,p = k cv A i − (cid:113) k (1 + cv A i ) − k cv A i cv A i ,µ = k − p E [ A i ] . If k >
2, this leads to E [ A i ] g i (0) = 0. 7he distributions in Examples 1 − E [ A i ] g i (0) can be computed if the density of the (scaled)interarrival times is known. If no information is available about the complete density, but thefirst two moments of A i are known, Whitt suggests to use the following approximation for E [ A i ] g i (0): E [ A i ] g i (0) = cv Ai cv Ai +1 if cv A i > , (cid:0) cv A i (cid:1) if cv A i ≤ , where cv A i is the squared coefficient of variation of the interarrival times of Q i . This ap-proximation is exact for cv A i >
1, if the interarrival time distribution is a hyper-exponentialdistribution as discussed in Example 2. For cv A i ≤
1, the approximation is rather arbitrary,but Example 3 shows that E [ A i ] g i (0) becomes small (or even zero) very rapidly as cv A i getssmaller.Summarising, the LT limit of a GI/G/ O ( ρ i ) terms and higher) is: E [ W LT i,GI/G/ ] = ρ i E [ A i ] g i (0) E [ B res i ] . (3.5)For Poisson arrivals ( E [ A i ] g i (0) = 1), it is known that E [ W i,M/G/ ] = ρ i − ρ i E [ B res i ] = ρ i E [ B res i ]+ O ( ρ i ), which is consistent with our approximation.The second step in determining the LT limit of the mean waiting time of a type i customerin a polling system, is finding the LT limits of E [ I res i ], the mean residual intervisit time of Q i , and (for gated service only) E [ V i I i ] E [ I i ] , the mean visit time V i given that it is observed fromthe following intervisit time I i . In this LT analysis we need to focus on first order terms only.Noting the fact that I i = S i + V i +1 + S i +1 + · · · + V i + N − + S i + N − , we condition on the momentat which I i is observed. We distinguish between two cases. The moment of observation eithertakes place during a visit time, or during a switch-over time: E [ I LT,res i ] = N − (cid:88) j =1 E [ V i + j ] E [ I i ] E [ I LT,res i | observed during V i + j ]+ N − (cid:88) j =0 E [ S i + j ] E [ I i ] E [ I LT,res i | observed during S i + j ] . (3.6) Observation during visit time.
The probability that a random observation epoch takesplace during a visit time, say V j , is E [ V j ] E [ I i ] , for any j (cid:54) = i . However, we are only interested inorder ρ terms, so this probability simplifies to E [ V j ] E [ I i ] = ρ j E [ C ](1 − ρ i ) E [ C ] = ρ j + O ( ρ ) . The fact that this probability is O ( ρ ), implies that all further O ( ρ ) terms can be ignored in E [ I LT,res i | observed during V j ], because in LT we focus on first order terms only.The length of the residual intervisit time is the length of the residual visit period of type j customers, V res j , plus all switch-over times S j + · · · + S i − , plus all visit times V j +1 + · · · + V i − .8he first term simplifies to E [ V res j ] = E [ B res j ] + O ( ρ ). The terms E [ V k | observed from V j ] , k = j + 1 , . . . , i −
1, in light traffic, are all O ( ρ ). Summarising, the mean residual intervisitperiod when observed during V j is simply a mean residual service time E [ B res j ], plus all meanswitch-over times E [ S j + · · · + S i − ], plus O ( ρ ) terms: E [ I LT,res i | observed during V j ] = E [ B res j ] + i − (cid:88) k = j E [ S k ] + O ( ρ ) . (3.7)The intuition behind this equation is that, in light traffic, the probability of having anotherservice during the residual cycle is negligible, i.e., O ( ρ ). Hence, the length of the residualintervisit time is solely determined by the residual service time and the remaining switch-overtimes in the cycle. Observation during switch-over time.
We continue by determining the mean residualintervisit period, conditioned on a random observation epoch during a switch-over time, say S j , j = 1 , . . . , N . The probability that such an epoch takes place during S j , is E [ S j ] E [ I i ] = E [ S j ](1 − ρ i ) E [ C ] = E [ S j ] E [ S ] 1 − ρ − ρ i = E [ S j ] E [ S ] (1 − ρ + ρ i ) + O ( ρ ) . It becomes apparent from this expression that things get slightly more complicated now,because order ρ terms in the conditional residual intervisit time may no longer be neglected.The residual intervisit time now consists of the residual switch-over time S res j , plus the switch-over times S j +1 + · · · + S i − , plus all visit periods V j +1 + · · · + V i − . The length of a visit period V k , for k > j , is the sum of the busy periods of all type k customers that have arrived during S k − N , . . . , S j − , S past j , S res j , and S j +1 , . . . , S k − . By S past j we denote the elapsed switch-overtime during which the intervisit period is observed, which has the same distribution as theresidual switch-over time S res j . Compared to an observation during a visit time, it is moredifficult to determine the conditional mean length of a busy period E [ V k | observed during S j ]under LT. We use a heuristic approach, which is exact if the arrival process of type k customersis Poisson, and approximate it by: E [ V k | observed during S j ] ≈ ρ k (cid:88) l (cid:54) = j E [ S l ] + E [ S past j ] + E [ S res j ] + O ( ρ ) , k = j +1 , . . . , i − . If A k is exponentially distributed, the above expression is exact. Nevertheless, numericalexperiments have shown that this approximative assumption has no or at least negligibleimpact on the accuracy of the approximated mean waiting times. Summarising: E [ I LT,res i | observed during S j ] ≈ j − (cid:88) k = i E [ S k ] (cid:0) i + N − (cid:88) l = j +1 ρ l (cid:1) + E (cid:16) S past j (cid:17) (cid:0) i + N − (cid:88) k = j +1 ρ k (cid:1) + E (cid:0) S res j (cid:1) (cid:0) i + N − (cid:88) k = j +1 ρ k (cid:1) + i + N − (cid:88) k = j +1 E [ S k ] (cid:0) i + N − (cid:88) l = j +1 ρ l (cid:1) + O ( ρ ) . (3.8)9he expression for I res i under light traffic conditions now follows from substituting (3.7) and(3.8) in (3.6). The result can be rewritten to: E [ I LT,res i ] ≈ i + N − (cid:88) j = i +1 ρ j E [ B res j ] + i + N − (cid:88) j = i +1 ρ j i + N − (cid:88) k = j E [ S k ]+ i + N − (cid:88) j = i E [ S ] E (cid:0) S j (cid:1) (cid:0) − ρ + ρ i + 2 i + N − (cid:88) k = j +1 ρ k (cid:1) + 1 E [ S ] i + N − (cid:88) j = i j − (cid:88) k = i E [ S j ] E [ S k ] (cid:0) i + N − (cid:88) l = j +1 ρ l (cid:1) + i + N − (cid:88) k = j +1 E [ S j ] E [ S k ] (cid:0) − ρ + ρ i + i + N − (cid:88) l = j +1 ρ l (cid:1) + O ( ρ )= i + N − (cid:88) j = i +1 ρ j E [ B res j ] + i + N − (cid:88) j = i +1 ρ j i + N − (cid:88) k = j E [ S k ]+ (1 − ρ + ρ i ) E [ S res ] + 1 E [ S ] i + N − (cid:88) j = i i + N − (cid:88) k = i E [ S j S k ] (cid:0) i + N − (cid:88) l = j +1 ρ l (cid:1) + O ( ρ )= (1 + ρ ) E [ S res ] + ρ E [ B res ] − ρ i E [ B res i ] + ρ i (cid:0) E [ S res ] − E [ S ] (cid:1) − E [ S ] N − (cid:88) j =0 j (cid:88) k =0 ρ i + k V ar[ S i + j ]+ O ( ρ ) , (3.9)for i = 1 , . . . , N . The last step in (3.9) follows after some straightforward (but tedious)rewriting.The Fuhrmann-Cooper decomposition of the mean waiting time for customers in a pollingsystem with gated service (3.3), also requires the computation of E [ V i I i ] E [ I i ] under LT conditions.Here, again, we have to resort to using a heuristic and use E [ V i I i ] E [ I i ] = ρ i E [ S ] + O ( ρ ), becausethis value is exact in the case of Poisson arrivals. This term is the mean length of the visittime V i given that it is observed during the following intervisit time I i . The term appearsbecause, contrary to exhaustive service, type i customers arriving during V i are not serveduntil the next cycle. However, it is easier to consider E [ V i I i ] / E [ V i ] instead, and to use therelation E [ V i I i ] E [ I i ] = E [ V i I i ] E [ V i ] × E [ V i ] E [ I i ] . The term E [ V i I i ] / E [ I i ] is the mean length of the intervisit time I i following V i , given that itis observed during this visit time V i . Firstly, we note that E [ V i ] E [ I i ] = ρ i − ρ i = ρ i + O ( ρ ) . This implies that we can ignore all O ( ρ ) terms in E [ V i I i ] / E [ V i ], which means that only theswitch-over times play a role, E [ V i I i ] E [ V i ] = E [ S ] + O ( ρ ) . E [ V i I i ] E [ I i ] = ρ i E [ S ] + O ( ρ ) , (3.10)in the case of Poisson arrivals. If the arrival process is not Poisson, this is not exact, but weuse it as an approximation.Having made all required preparations, we are ready to formulate the main result of thepresent subsection. Under light traffic, an approximation for the mean waiting time of a type i customer in a polling model with general arrivals and respectively exhaustive and gatedservice in Q i , is: E [ W LT,exh i ] ≈ E [ S res ] + ρ i ( E [ ˆ A i ]ˆ g i (0) − E [ B res i ] + ρ E [ B res ] + ( ρ − ρ i ) ( E [ S ] − E [ S res ])+ 1 E [ S ] i + N − (cid:88) k = i +1 ρ k k − (cid:88) j = i V ar[ S j ] + O ( ρ ) , i = 1 , . . . , N, (3.11) E [ W LT,gated i ] ≈ E [ W LT,exh i ] + ρ i E [ S ] , (3.12)where ˆ g i ( t ) is the density of the interarrival times of type i customers at ρ = 1. Equation(3.11) follows from substitution of (3.5) and (3.9) in E [ W i ] ≈ E [ W i,GI/G/ ] + E [ I res i ] , i = 1 , . . . , N . (3.13)For Poisson arrivals, (3.11) and (3.12) are exact. The LT limit for polling systems withBernoulli service (and Poisson arrivals) has been experimentally found in [2] and, indeed, itcan be shown that their result for exhaustive service, which is a special case of Bernoulliservice, agrees with our result after substituting E [ ˆ A i ]ˆ g i (0) = 1 in (3.11). Heavy traffic limits in polling systems have been studied by Coffman et al. [5, 6], and by Olsenand Van der Mei [16, 17]. In these papers, the HT limits of the waiting time distributionsare found under the assumption of Poisson arrivals. For general renewal arrivals, a proof isgiven for the special case N = 2 (cf. [5, 6]), and a strong conjecture for larger values of N (cf.[17]). In [23], the following result for the mean waiting time is proven rigorously for pollingsystems with renewal arrivals: E [ W HT i ] = ω i − ρ + o ((1 − ρ ) − ) , ρ ↑ . (3.14)Obviously, in HT, all queues become unstable and, thus, E [ W i ] tends to infinity for all i .The rate at which E [ W i ] tends to infinity as ρ ↑ ω i , which is referred toas the mean asymptotic scaled delay at queue i , and depends on the service discipline. Forexhaustive service, ω i = 1 − ˆ ρ i (cid:32) σ (cid:80) Nj =1 ˆ ρ j (1 − ˆ ρ j ) + E [ S ] (cid:33) , i = 1 , . . . , N, with σ := N (cid:88) i =1 ˆ λ i (cid:16) V ar[ B i ] + ˆ ρ i V ar[ ˆ A i ] (cid:17) . A i ( i =1 , . . . , N ) fixed up to a common scaling constant ρ . Notice that in the case of Poisson arrivalswe have σ = E [ B ] / E [ B ].For gated service, we have ω i = 1 + ˆ ρ i (cid:32) σ (cid:80) Nj =1 ˆ ρ j (1 + ˆ ρ j ) + E [ S ] (cid:33) . Now that we have the expressions for the mean delay in both LT and HT, we can determinethe constants K ,i , K ,i , and K ,i in approximation formula (2.1). We simply impose therequirements that approximation (2.1) results in the same mean waiting time for ρ = 0 as theLT limit, and for ρ ↑ ρ terms, we also add the requirement that the derivativewith respect to ρ , taken at ρ = 0, of our approximation is equal to the derivative of the LTlimit. A more formal definition of these requirements is presented below: E [ W i, app ] (cid:12)(cid:12) ρ =0 = E [ W i ] (cid:12)(cid:12) ρ =0 , dd ρ E [ W i, app ] (cid:12)(cid:12) ρ =0 = dd ρ E [ W i ] (cid:12)(cid:12) ρ =0 , (1 − ρ ) E [ W i, app ] (cid:12)(cid:12) ρ =1 = (1 − ρ ) E [ W i ] (cid:12)(cid:12) ρ =1 . This leads to (2.1) as approximation for E [ W i ] in a polling system with general arrivals.Constants K ,i , K ,i , and K ,i are defined in (2.2)–(2.4) for systems with exhaustive service,or (2.5)–(2.7) for gated service. A desirable property of an approximation is that it matches known exact results. In thepresent section we discuss several cases where the interpolation approximation yields exact results. Most cases require Poisson arrivals, but it is shown that also in two limiting caseswhere exact results are available for general arrivals, i.e., heavy traffic and large switch-overtimes, the approximation is exact. It further turns out that, in case of Poisson arrivals,the approximated mean waiting times satisfy the pseudo-conservation law , implying that theweighted sum (cid:80) Ni =1 ρ i E [ W i, app ] is exact for each load 0 ≤ ρ <
1. In fact, this appears tobe true for any interpolation approximation for the mean waiting times of the form (3.1)matching the HT limit and the LT limits of order 0 up to order k , provided k >
0. Theseproperties of the interpolation approximation indicate that it is the “natural” approximation,given the HT and LT limits.
Light and heavy traffic.
The light traffic limit of E [ W i ], given by (3.11) for exhaustiveservice and by (3.12) for gated service, is exact for Poisson arrivals. The heavy traffic limit(3.14) of E [ W i ] is even exact for renewal arrivals. An appropriate choice of constants K ,i , K ,i K ,i can reduce (2.1) to either (3.11), (3.12), or (3.14). Since the LT and HT limits havebeen used in the set of equations that determine the coefficients of the approximation, it goeswithout saying that E [ W i, app ] is equal to (3.11) (or (3.12) for gated service) and (3.14), for ρ ↓ ρ ↑ Symmetric system.
If ˆ ρ i = N for all i = 1 , . . . , N , all B i have the same distribution, andthe variances V ar[ S i ] of all switch-over times are equal, then our approximation is exact if allinterarrival distributions are exponential. For exhaustive service, we obtain K ,i = E [ B res ] + N − N E [ S ] − (cid:18) − N (cid:19) E [ S res ] + 1 E [ S ] i + N − (cid:88) k = i +1 ˆ ρ k k − (cid:88) j = i V ar[ S j ]= E [ B res ] + N − N E [ S ] − (cid:18) − N (cid:19) E [ S res ] + N − N V ar[ S ]2 E [ S ]= E [ B res ] + (cid:18) − N (cid:19) E [ S ]2 − E [ S res ] , and K ,i = 0. This implies that E [ W i, app ] = E [ W i, symm ], since E [ W i, symm ] = ρ − ρ E [ B res ] + E [ S res ] + ρ (1 − N )1 − ρ E [ S ]2 . Note that E [ W i, symm ] is of the form (3.1) with q ( ρ ) being a linear polynomial. In fact, thisimmediately implies that the interpolation is exact, given the (exact) LT and HT limits.Further, we do not require that the mean switch-over times E [ S i ] are equal. One can verifythat the same holds for gated service. Single queue (vacation model).
An immediate consequence of the fact that our approx-imation is exact in symmetric polling systems with Poisson arrivals, is that it also gives exactresults for the mean waiting time of customers in a single-queue polling system with Poissonarrivals. A polling system consisting of only one queue, but with a switch-over time betweensuccessive visits to this queue, is generally referred to as a queueing system with multipleserver vacations.
Large switch-over times.
For S deterministic, S → ∞ , and, again, under the assumptionof Poisson arrivals, it is proven in [27, 28] that E [ W i ] S → − ρ i − ρ ) for exhaustive service. It caneasily be verified that our approximation has the same limiting behaviour:lim S →∞ E [ W i, app ] S = 1 − ρ i − ρ ) . (3.15)For gated service, E [ W i, app ] S → ρ i − ρ ) , which is also the exact limit (see, e.g., [26]).Olsen [15] studies the effects of large switch-over times in polling systems operating underHT conditions. She discovers that, under these conditions, the limiting behaviour (3.15) isalso exhibited in polling systems with general renewal arrivals .13 iscellaneous other exact results. The approximation is also exact in several othercases, all with Poisson arrivals, when the parameter values are carefully chosen. The re-lations between the input parameters that yield exact approximation results become verycomplicated, especially in polling systems with more than two queues. We only mentionone interesting example here: our approximation gives exact results for a two-queue pollingsystem with exhaustive service and E [ B ] = E [ B ] , E [ S ] = E [ S ] , cv A = cv A , cv B = cv B , cv S = cv S , (3.16)if the following constraint is satisfied: ρ = 1 + I A i I A i − cv S i cv B i · E [ S i ] E [ B i ] , (3.17)where I A i = ˆ ρ ˆ ρ is the ratio of the loads of the two queues. Obviously, if I A i = 1, the systemis symmetric and our approximation gives exact results regardless of the other parametersettings. Pseudo-conservation law.
A well-known result in polling literature, is the so-called pseudo-conservation law , derived by Boxma and Groenendijk [4] using the concept of work decompo-sition. This law gives the following exact expression for the weighted sum of the mean waitingtimes in a polling system with Poisson arrivals: N (cid:88) i =1 ρ i E [ W i ] = ρ − ρ E [ B res ] + ρ E [ S res ] + E [ S ]2 ρ − (cid:80) Nj =1 ρ j − ρ + N (cid:88) j =1 E [ Z jj ] , (3.18)where E [ Z jj ] denotes the mean amount of work left behind in Q j at the completion of avisit of the server to Q j . It is shown in [4] that E [ Z jj ] is the only term that depends onthe service discipline. For exhaustive service E [ Z jj ] = 0, for gated service E [ Z jj ] = ρ j E [ S ]1 − ρ .It can be shown that the interpolation approximation (2.1) satisfies the pseudo-conservationlaw in the case of Poisson arrivals: if E [ ˆ A i ]ˆ g i (0) = 1 for i = 1 , . . . , N , then (cid:80) Ni =1 ρ i E [ W i, app ]equals the right-hand side of (3.18). In fact, in case of Poisson arrivals, any interpolationapproximation for the mean waiting times of the form (3.1), matching the HT limit and theLT limits of order 0 up to order k , satisfies (3.18), provided k >
0. To establish this result,we first rewrite (3.18) in the form Q ( ρ ) = 0, by moving the terms at right-hand side of (3.18)to the left. Now let ˜ Q ( ρ ) denote the version of Q ( ρ ) where the mean waiting times have beenreplaced by their interpolation approximation. Clearly, the interpolation approximation andthe right-hand side of (3.18) are both of the form (3.1), and thus ˜ Q ( ρ ) is also of the form˜ Q ( ρ ) = ˜ q ( ρ )1 − ρ , where ˜ q ( ρ ) is a polynomial of degree k + 2. Since the interpolation approximation matchesthe LT limits of order 0 up to order k , it follows that ˜ Q ( n ) (0) = Q ( n ) (0) = 0, and hence,˜ q ( n ) (0) = 0 for all n = 0 , , . . . , k + 1. Further, by the HT limit, ˜ q (1) = lim ρ → (1 − ρ ) ˜ Q ( ρ ) =lim ρ → (1 − ρ ) Q ( ρ ) = 0. This implies that ˜ q ( ρ ) = 0, and thus ˜ Q ( ρ ) = 0 for all ρ .14 Numerical study
Before we study the accuracy of the approximation to a large test bed of polling systems, wejust pick a rather arbitrary, simple system to compare the approximation with exact resultsin order to get some initial insights. Consider a three-queue polling system with loads of Q , Q , and Q divided as follows: ˆ ρ = 0 . , ˆ ρ = 0 .
3, and ˆ ρ = 0 .
6. All service times andswitch-over times are exponentially distributed, with mean 1. The interarrival times haveSCV cv A i = 3 for i = 1 , ,
3. In Figure 1 we plot the approximated mean waiting time of Q , E [ W , app ], versus the load of the system ρ . Since this system cannot be analysed analytically,we compare the approximated values with simulated values. Both in the approximation andin the simulation we fit a H distribution as described in Example 2.The errors are largest for Q , which is the reason why we chose this queue in particular inFigure 1. The most important information that this figure reveals, is that even though theaccuracy of the approximation is worst for this queue (a relative error of − .
47% for ρ = 0 . Q and Q are 3 .
10% and 2 .
90% respectively.In order to get more insight in the numerical accuracy of the approximation for a wide varietyof different parameter settings, we create a large test bed in the next subsection and comparethe approximation with exact or simulated results. It turns out that the maximum relativeerrors for most of the polling systems are smaller than the one selected in the above example. Ρ E H W L SimulationApproximation
Figure 1: Approximated and simulated mean waiting time E [ W ] of Q of the example insubsection 4.1. 15 .2 Accuracy of the approximation In the present section we study the accuracy of our approximation. We compare the approx-imated mean waiting times of customers in various polling systems to the exact values. Thecomplete test bed of polling systems that are analysed, contains 2304 different combinationsof parameter values, all listed in Table 1. We show detailed results for exhaustive servicefirst, and discuss polling systems with gated service at the end of this section. We have variedParameter Notation ValuesNumber of queues N , , , ρ . , . , . , . , . , . cv A i . , , cv B i . , cv S i . , I A i , I B i , I S i /B i , . . .
2, and included ρ = 0 .
99 to analyse the limitingbehaviour of our approximation when the load tends to 1. The SCV of the interarrival times, cv A i , is varied between 0 .
25 and 2. In case of non-Poisson arrivals, i.e. cv A i (cid:54) = 1, the exactvalues have been established through extensive simulation because they cannot be obtained inan analytic way. In these simulations we fit a phase-type distribution to the first two momentsof the interarrival times, as described in Examples 2 and 3. For service times and switch-overtimes, only SCVs of 0 .
25 and 1 are considered. SCVs greater than 1 are less common inpractice and are discussed separately from the test bed later in this section. The imbalancein interarrival times and service times, I A i and I B i , is the ratio between the largest and thesmallest mean interarrival/service time. The interarrival times are determined in such a way,that the overall mean is always 1, λ is the largest and λ N the smallest, and the steps betweenthe λ i are linear. E.g., for N = 5 and I A i = 5 we get λ i = 2 − i/ , i = 1 , . . . ,
5. The meanservice times E [ B i ] increase linearly in i = 1 , . . . , N , with E [ B N ] = I B i E [ B ] (so E [ B ] is thesmallest mean service time). They follow from the relation (cid:80) Ni =1 λ i E [ B i ] = ρ . E.g., for N = 5,and I A i = I B i = 5 we get E [ B i ] /ρ = 3 i/
35. The last parameter that is varied in the test bed,is the ratio between the mean switch-over times and the mean service times, I S i /B i = E [ S i ] E [ B i ] .The total number of systems analysed is 4 × × × = 2304. A system consisting of N queues results N mean waiting times, E [ W ] , . . . , E [ W N ], so in total these 2304 systems yield8064 mean waiting times. The absolute relative errors, defined as | o − e | /e , where o standsfor observed (approximated) value, and e stands for expected (exact) value, are computed forall these 8064 queues. Table 2 shows these relative errors (times 100%) categorised in binsof 5%. In this table, and in all other tables, results for systems with a different number ofqueues are displayed in separate rows. The reader should keep in mind that the statisticsin each row are based on × × N absolute relative errors, where N is the number ofqueues used in the specified row. Table 2 shows that, e.g., 98 .
84% of the approximated meanwaiting times in polling systems consisting of 3 queues deviate less than 5% from their true16alues. From Table 2 it can be concluded that the approximation accuracy increases with thenumber of queues in a polling system. More specifically, for systems with more than 2 queues,no approximation errors are greater than 10%, and the vast majority is less than 5%. Themean relative errors for N = 2 , . . . , . , . , .
70% and 0 . ρ is varied, and it can beseen that for a load of ρ = 0 . ρ = 0 . . cv A i = 1. In Table 3(c) the impact of imbalance in a polling systemon the accuracy is depicted, and, as could be expected, it can be concluded that a highimbalance in either service or interarrival times has a considerable, negative, impact on theapproximation accuracy. Polling systems with more than 2 queues are much less bothered bythis imbalance than polling systems with only 2 queues. N −
5% 5 −
10% 10 −
15% 15 − .
46 10 .
24 2 .
78 0 .
523 98 .
84 1 .
16 0 .
00 0 .
004 99 .
78 0 .
22 0 .
00 0 .
005 99 .
93 0 .
07 0 .
00 0 . In this subsection we discuss several cases that are left out of the test bed because they mightnot give any new insights, or because the combination of parameter values might be rarelyfound in practice.
More queues.
Firstly, we discuss polling systems with more than 5 queues briefly. Withoutlisting the actual results, we mention here that the approximations become more and moreaccurate when letting N grow larger, and still varying the other parameters in the same wayas is described in Table 1. For N = 10 already, all relative errors are less than 5%, with anaverage of less than 0 .
5% and it only gets smaller as N grows further. More variation in service times and switch-over times.
In the test bed we only useSCVs 0 .
25 and 1 for the service times and switch-over times, because these seem more relevant17
Load ( ρ )0 .
10 0 .
30 0 .
50 0 .
70 0 .
90 0 .
992 0 .
31 1 .
81 3 .
41 4 .
17 2 .
70 0 .
673 0 .
16 0 .
84 1 .
44 1 .
69 1 .
07 0 .
394 0 .
13 0 .
68 1 .
14 1 .
28 0 .
73 0 .
255 0 .
11 0 .
57 0 .
94 1 .
03 0 .
57 0 . N SCV interarrival times ( cv A i )0 .
25 1 22 2 .
27 1 .
76 2 .
503 1 .
36 0 .
52 0 .
924 1 .
13 0 .
29 0 .
695 0 .
97 0 .
19 0 . N Imbalance interarrival and service times I A i = 1 , I B i = 1 I A i = 1 , I B i = 5 I A i = 5 , I B i = 1 I A i = 5 , I B i = 52 0 .
69 2 .
92 2 .
80 2 .
303 0 .
65 1 .
27 0 .
75 1 .
064 0 .
56 0 .
89 0 .
62 0 .
735 0 .
49 0 .
69 0 .
53 0 . N ) and totalload of the system (a), SCV interarrival times (b), and imbalance of the interarrival andservice times (c).from a practical point of view. As the coefficient of variation grows larger, our approximationwill become less accurate. E.g., for Poisson arrivals we took cv B i ∈ { , } , cv S i ∈ { , } ,and varied the other parameters as in our test bed (see Table 1). This way we reproducedTable 2. The result is shown in Table 4 and indicates that the quality of our approximationdeteriorates in these extreme cases. The mean relative errors for N = 2 , . . . , . . . . N −
5% 5 −
10% 10 −
15% 15 − .
22 14 .
84 6 .
51 2 .
083 89 .
76 7 .
29 2 .
08 0 .
694 94 .
53 4 .
56 0 .
91 0 .
005 97 .
71 2 .
19 0 .
10 0 . mall switch-over times. Systems with small switch-over times, in particular smallerthan the mean service times, also show a deterioration of approximation accuracy - especiallyin systems with 2 queues. In Figure 2 we show an extreme case with N = 2, service timesand switch-over times are exponentially distributed with E [ B i ] = and E [ S i ] = for i = 1 ,
2, which makes the mean switch-over times 5 times smaller than the mean servicetimes. Furthermore, the interarrival times are exponentially distributed with λ = 5 λ . InFigure 2 the mean waiting times of customers in both queues are plotted versus the loadof the system. Both the approximation and the exact values are plotted. For customers in Q the mean waiting time approximations underestimate the true values, which leads to amaximum relative error of − .
2% for ρ = 0 . E [ W , app ] = 0 .
43, whereas E [ W ] = 0 . Q is systematically overestimating thetrue value. The maximum relative error is attained at ρ = 0 . .
8% ( E [ W , app ] = 0 . E [ W ] = 0 . sojourn time is already much better approximated. Nevertheless,this example illustrates one of the situations where our approximation gives unsatisfactoryresults. Ρ E H W L ExactApproximation Ρ E H W L Figure 2: Approximated and exact mean waiting times for a two-queue polling system withsmall switch-over times.
The only alternative approximations that exist for polling systems with non-exponential inter-arrival times, only perform well under extreme, limiting conditions. In [17, 23] it is suggestedto use the HT limit (3.14) as an approximation, but the accuracy is only found to be accept-able for ρ > .
8. Another approximation for the mean waiting time in polling systems withnon-exponential interarrival times uses the limit for S → ∞ [27, 28]. This approximation isusable if either the total switch-over time in the system is large and the switch-over times havelow variance, or if the total switch-over time in the system is large and the system is in heavytraffic. For completeness, we mention that it is also possible to construct an approximationthat is purely based on the LT limit, developed in the present paper: E [ W LT i ] ≈ K ,i + ( K ,i − K ,i ) ρ − ρ . (4.1)We do not wish to go into further details on this topic, because the accuracy of (4.1) turnsout to be worse for high loads. This makes our approximation, which is exact in all these19imiting cases, the only one which can be applied under all circumstances. We support thisstatement by reproducing Table 3(a) for the three alternative approximations, based on HTlimit, large switch-over times, and LT limit. A comparison of Table 5, which displays theseresults, to Table 3(a), clearly indicates the drawbacks of using approximations based on onelimiting case only. N Load ( ρ )0 .
10 0 .
30 0 .
50 0 .
70 0 .
90 0 .
992 58 .
97 51 .
03 40 .
34 27 .
47 11 .
34 1 .
463 30 .
62 25 .
66 19 .
63 12 .
61 4 .
56 0 .
624 22 .
49 18 .
61 14 .
02 8 .
80 3 .
08 0 .
425 18 .
06 14 .
83 11 .
10 6 .
88 2 .
36 0 . N Load ( ρ )0 .
10 0 .
30 0 .
50 0 .
70 0 .
90 0 .
992 27 .
13 31 .
08 35 .
95 41 .
32 46 .
78 49 .
233 19 .
28 21 .
68 24 .
51 27 .
52 30 .
33 31 .
444 15 .
05 16 .
75 18 .
76 20 .
92 22 .
93 23 .
675 12 .
37 13 .
67 15 .
24 16 .
96 18 .
56 19 . N Load ( ρ )0 .
10 0 .
30 0 .
50 0 .
70 0 .
90 0 .
992 0 .
34 2 .
65 7 .
00 13 .
42 21 .
90 26 .
823 0 .
24 1 .
68 3 .
90 6 .
70 9 .
62 10 .
824 0 .
19 1 .
31 2 .
97 4 .
87 6 .
75 7 .
465 0 .
16 1 .
09 2 .
45 4 .
02 5 .
54 6 . N ) and totalload of the system, for three alternative approximations: based on the HT limit (a), basedon large switch-over times (b), and based on the LT limit (c).For polling systems with Poisson arrivals, several alternative approximations have been de-veloped in existing literature. The best one among them (see, e.g., [3, 7, 11]) uses the relation E [ W i ] = (1 ± ρ i ) E [ C res i ], where C i is the cycle time, starting at a visit completion to Q i whenservice is exhaustive, and starting at a visit beginning for gated service. By ± we mean − for exhaustive service, and + for gated service. The mean residual cycle time, E [ C res i ], isassumed to be equal for all queues, i.e. E [ C res i ] ≈ E [ C res ], and can be found by substituting E [ W i ] ≈ (1 ± ρ i ) E [ C res ] in the pseudo-conservation law (3.18). We have used this PCL-basedapproximation to estimate the mean waiting times of all queues in the test bed describedin Table 1, but taking only the 768 cases where cv A i = 1. Table 6 shows the mean relative20rrors for our approximation (a) and the PCL approximation (b), categorised in bins of 5%as was done before in Table 2. From these tables (and from other performed experimentsthat are not mentioned for the sake of brevity) it can be concluded that for N > ρ , and the PCL approximation being slightly better for high values of ρ (bothmethods are asymptotically exact as ρ ↑ N = 2 our method suffers greatlyfrom imbalance in the system, whereas the PCL approximation proves to be more robust. N −
5% 5 −
10% 10 − .
32 9 .
11 1 .
563 100 .
00 0 .
00 0 .
004 100 .
00 0 .
00 0 .
005 100 .
00 0 .
00 0 . N −
5% 5 −
10% 10 − .
09 2 .
86 1 .
043 99 .
31 0 .
69 0 .
004 100 .
00 0 .
00 0 .
005 100 .
00 0 .
00 0 . Until now we have only shown and discussed approximation results for polling systems withexhaustive service. The complete test bed described in Table 1 has also been analysed forpolling systems where each queue receives gated service. As can be seen in Table 7, the overallquality of the approximation is good, but worse than for polling systems with exhaustiveservice. More details on the reason for these inaccuracies can be found in Table 8, which isthe equivalent of Table 3 for gated service. Table 8(b) illustrates that there is now a hugedifference between systems with Poisson arrivals, and systems with non-Poisson arrivals. Forthe cases with cv A i = 1, the approximation is extremely accurate, even for two-queue pollingsystems. The accuracy in cases with cv A i (cid:54) = 1 is worse, which is caused by the assumptionsthat are made to approximate the LT limit (3.12). Firstly, the decomposition (3.3) does nothold for non-Poisson arrivals, and secondly, the terms E [ I res i ] and E [ V i I i ] E [ I i ] in this decompositionhave only been approximated. For exhaustive service, these assumptions do not have muchnegative impact on the accuracy, but apparently, for gated service, they do. The meanrelative errors for N = 2 , . . . , . . . . N = 2 , . . . , . , . , . . −
5% 5 −
10% 10 −
15% 15 − .
55 12 .
33 2 .
95 1 .
563 85 .
42 10 .
53 3 .
13 0 .
814 88 .
85 8 .
46 2 .
43 0 .
265 92 .
22 6 .
60 1 .
15 0 . N Load ( ρ )0 .
10 0 .
30 0 .
50 0 .
70 0 .
90 0 .
992 2 .
64 4 .
55 4 .
31 3 .
10 1 .
25 0 .
373 2 .
03 3 .
78 3 .
68 2 .
68 1 .
04 0 .
304 1 .
62 3 .
14 3 .
13 2 .
32 0 .
92 0 .
285 1 .
35 2 .
67 2 .
71 2 .
03 0 .
81 0 . N SCV interarrival times ( cv A i )0 .
25 1 22 4 .
72 0 .
34 3 .
053 4 .
06 0 .
17 2 .
534 3 .
45 0 .
10 2 .
165 2 .
98 0 .
08 1 . N Imbalance interarrival and service times I A i = 1 , I B i = 1 I A i = 1 , I B i = 5 I A i = 5 , I B i = 1 I A i = 5 , I B i = 52 2 .
76 2 .
64 2 .
81 2 .
593 2 .
28 2 .
25 2 .
27 2 .
214 1 .
93 1 .
91 1 .
90 1 .
875 1 .
64 1 .
66 1 .
64 1 . N ) and total load of the system (a), SCV interarrival times (b), and imbalance ofthe interarrival and service times (c). The research that is done in the present paper can be extended in many different directions.In this section we discuss some possibilities that we find most relevant.
Higher moments.
Firstly, a logical follow-up step would be to use the same approachto find approximations for higher moments of the waiting time distribution as well. Thismight prove to be a hard exercise, since the LT limit of E [ W i ] is unknown and, although itsderivation might follow the same lines as in Section 3, it probably requires substantially moreeffort. In [13], explicit expressions for the second moments of the waiting time distributionsare given, but only for symmetric systems with N = 2 , , and 4, and under the assumption ofPoisson arrivals. Also, the HT limit of E [ W i ] is unknown, although some research in this area22as already been done and in [17] a strong conjecture is given for the limiting distribution of W i as ρ ↑ E [ W i ]. Other service disciplines.
In the present paper, only exhaustive and gated service arediscussed. In order to obtain results for polling systems with some queues receiving exhaustiveservice, and others receiving gated service, only minor modifications should be made, but weleave this to the reader. It would be more challenging to generalise the approximation toa wider variety of service disciplines. In particular, it would be nice to have one expressionfor the mean waiting time of customers in a queue with an arbitrary branching-type servicediscipline (cf. [19]). The exhaustiveness of a branching-type service discipline (cf. [26]) mightappear in this expression. Gated and exhaustive are both branching type service disciplines,but are discussed separately in the present paper. The HT limit can most likely be establishedfor arbitrary branching type service disciplines (see conjectures in [17]), so the question thatremains is whether the LT limit can be found in a similar way.
Optimisation.
One of the main reasons to choose (2.1) as form of the interpolation, besidesits asymptotic correctness, is its simplicity. Having this exact and simple expression forthe approximate mean waiting times, makes it very useful for optimisation purposes. Inproduction environments, one can, for example, determine what the optimal strategy is tocombine orders of different types (i.e., determine what queue customers should join). Becausegeneral arrivals are supported, one can determine optimal sizes of batches in which items aregrouped and sent to a specific machine. The simplicity of (2.1) makes it possible for a managerto create a handy Excel sheet that can be used by operators to compute all kind of optimalparameter settings. No difficult computations are required at all, so a large variety of userscan use the approximation.In the present paper the accuracy of the approximation has been investigated and has beenfound to be very good in most situations. Another advantage of our approximation regardingoptimisation purposes, is that the general shape of the approximated curve follows the exactcurve very closely. Even in cases where the relative errors are rather large, like in Figure1, the shape of the actual curves is still very well approximated. This means that pluggingour approximation, instead of an exact expression if it had been available, in an optimisationfunction yields an optimum that should be close to the true optimum.
Polling Table.
The interpolation based approximation can also be extended to polling sys-tems where the visiting order of the queues is not cyclic. Waiting times in polling systems withso-called polling tables can be obtained in the same way as shown in the present paper. Boththe LT and HT limits are not difficult to determine in this situation, and the interpolationfollows directly from these limits.
Model.
The form of the interpolation might be changed to improve the accuracy of ap-proximations for cases that give less satisfactory results in the present form. E.g., one couldtry other functions than a second-order polynomial as numerator of (2.1). Alternatively, one23ould try to find a correction term which could be added to (2.1) to obtain better resultsfor, e.g., two-queue polling systems. But most of all, if an exact
LT limit of the mean wait-ing time in a polling system with non-Poisson arrivals could be found, the accuracy of theapproximation in the case of gated service might be improved.
Acknowledgements
The authors wish to thank Onno Boxma for valuable discussions and for useful comments onearlier drafts of the present paper.
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