aa r X i v : . [ m a t h . P R ] A ug On open problems in polling systems ∗ Marko Boon † [email protected] O.J. Boxma † [email protected] E.M.M. Winands ‡ [email protected], 2011 Abstract
In the present paper we address two open problems concerning polling systems, viz.,queueing systems consisting of multiple queues attended by a single server that visits thequeues one at a time. The first open problem deals with a system consisting of two queues,one of which has gated service, while the other receives 1-limited service. The second openproblem concerns polling systems with general (renewal) arrivals and deterministic switch-over times that become infinitely large. We discuss related, known results for both problems,and the difficulties encountered when trying to solve them.
Keywords:
Polling, gated, one-limited, branching-class, switch-over time asymptotics
A polling system is a queueing system consisting of multiple queues attended by a single server thatvisits the queues one at a time. Polling systems naturally arise in a large number of applicationareas, like • maintenance: a patrolling repairman visits various sites; • manufacturing: a machine successively produces items of various types; • computer-communication systems: a central computer cyclically polls the terminals on acommon link to inquire whether they have any data to transmit; • road traffic: traffic lights determine which traffic streams may proceed.In many of these applications, the server incurs a non-negligible switch-over time when switchingbetween queues.There is a huge body of literature on polling systems, in which the basic cyclic polling system andmany enhancements have been studied. Extensive surveys on polling systems and their applica-tions may be found in [13, 20, 24]. In this note we present two challenging open problems motivatedby two of the aforementioned application areas. By doing so, we want to stimulate research and newcollaborations in these directions. The first problem is motivated by a computer-communicationsystem application and seems to lead to a boundary value problem with a rather complicated shift.The second problem requires an asymptotic analysis of the waiting-time distribution and stemsfrom a manufacturing application. ∗ 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 Amsterdam,The Netherlands Model and notation
In the present paper (and in almost the whole polling literature), the server visits the N queues incyclic order Q , Q , . . . , Q N , Q , . . . , and the arrival processes of customers at the various queuesare assumed to be independent Poisson processes, with rate λ i at Q i , i = 1 , . . . , N . The servicerequirements at Q i , denoted by B i , are independent, identically distributed random variables withLST (Laplace-Stieltjes transform) β i ( · ), i = 1 , . . . , N . Similarly, the switch-over times between Q i and Q i +1 , denoted by S i , are independent, identically distributed with LST σ i ( · ), i = 1 , . . . , N .The sum of the switch-over times is denoted by S . We furthermore assume that all arrival, serviceand switch-over processes are independent, and that the various parameters are such that thejoint steady-state queue-length distributions at server visit epochs, server departure epochs andarbitrary epochs exist. We also introduce the notation ρ i = λ i E [ B i ], and ρ = P Ni =1 ρ i .A key aspect of polling systems is the service discipline at each queue. The three most importantservice disciplines are exhaustive (E): a queues is served until it is empty; gated (G): the serveronly serves those customers which were present at the start of the visit; and (1-L): theserver serves only one customer – if any is present. In a seminal paper, Resing [18] has shown thatthe PGF (Probability Generating Function) of the joint steady-state queue-length distribution atepochs at which the server arrives at, say, Q can be obtained explicitly for those polling systemsin which the service discipline at each queue is of a branching-type, viz., the following holds for i = 1 , . . . , N : If there are k i customers present at Q i at the start of a visit, then during the course of the visit,each of these k i customers will effectively be replaced in an i.i.d. manner by a random populationhaving PGF h i ( z , . . . , z N ) , which may be any N -dimensional PGF. The joint queue-length process at visit epochs then becomes an N -class branching process withimmigration (the immigration corresponding to arrivals during switch-over times). One may easilyverify that exhaustive and gated are branching-type disciplines, whereas 1-limited is not. In thegated case, h i ( z , . . . , z N ) = β i ( P Nj =1 λ j (1 − z j )), and in the exhaustive case, h i ( z , . . . , z N ) = π i ( P Nj = i λ j (1 − z j )), where π i ( · ) denotes the LST of the busy period distribution at Q i , whenviewed as an M/G/ G i ( z , . . . , z N ) of the joint queue-length distribution at the end of a visitto Q i and the PGF F i ( z , . . . , z N ) of the joint queue-length distribution at the start of that visit: G i ( z , . . . , z N ) = F i ( z , . . . , z i − , h i ( z , . . . , z N ) , z i +1 , . . . , z N ) . (1)Moreover, it is easily seen that F i +1 ( z , . . . , z N ) = σ i (cid:0) N X j =1 λ j (1 − z j ) (cid:1) G i ( z , . . . , z N ) . (2)Successively applying each of these equations once for i = 1 , . . . , N , one may now express F ( z , . . . , z N )into itself. After iteration this yields an expression for F ( z , . . . , z N ) in the form of an infinitesum of products.In the next two sections we formulate two open problems for polling systems with a partial,respectively full, branching-type service discipline. In this section we restrict ourselves to the case of N = 2 queues. We are interested in determining F ( z , z ) and F ( z , z ). After briefly discussing known results in Subsection 3.1, we formulate inSubsection 3.2 an open problem regarding the polling model with a gated and a 1-limited queue.2 .1 Known results for two-queue polling systems The polling models with discipline E/E (Exhaustive/Exhaustive), G/G and E/G fall in the classof multi-type branching, and are easily solved; E/E was already solved by Tak´acs [19] in 1968.We refer to the survey [20] and to [18] for the other cases and for extensions to a general numberof queues N . The 1-L/1-L model was solved by using the theory of boundary value problems; see[6] for the case of zero switch-over times, and [5] for the case of non-zero switch-over times. Nowlet us turn to E/1-L and G/1-L. First observe that, with Q having 1-L: G ( z , z ) = β ( λ (1 − z ) + λ (1 − z )) z [ F ( z , z ) − F ( z , F ( z , . (3)Combination of (1), (2) and (3) yields, after having introduced β i ( z , z ) := β i ( λ (1 − z ) + λ (1 − z )) and σ i ( z , z ) := σ i ( λ (1 − z ) + λ (1 − z )), i = 1 , F ( z , z ) = β ( z , z ) σ ( z , z ) z [ σ ( z , z ) F ( h ( z , z ) , z ) − σ ( z , F ( h ( z , , σ ( z , z ) σ ( z , F ( h ( z , , . (4)The E/1-L model with zero switch-over times is simply a two-class nonpreemptive priority model.Ibe [10] considers the case of non-zero switch-over times, obtaining the marginal queue-lengthdistribution in Q at polling instants of that queue. It is less well-known that the joint queue-length distributions at polling instants of a queue can also be found in a quite straightforwardmanner. This is accomplished by substituting h ( z , z ) = π ( λ (1 − z )) into (4), calling thisfunction g ( z ), and observing that F ( h ( z , ,
0) = F ( g (0) ,
0) is a constant, say C , not dependingon z : F ( z , z ) = β ( z , z ) σ ( z , z ) z [ σ ( z , z ) F ( g ( z ) , z ) − Cσ ( z , Cσ ( z , z ) σ ( z , . (5)The substitution z = g ( z ) finally solves the problem. Details may be found in Section 6.3 ofthe PhD thesis of Groenendijk [9]. E/1-L appears to be conceptually easier than E/E or anyother known polling model, not requiring a branching-type sum-of-infinite-products solution, andneither the solution of a boundary value problem. Remark 3.1.
Our sketch of the analysis of E/1-L reveals that one can extend that analysis tothe case in which h ( z , z ), which above equals π ( λ (1 − z )), is some arbitrary PGF g ( z ). Forexample, when the server is at Q , one could have Poisson arrivals with rate λ ∗ , or batch Poissonarrivals, at Q . Remark 3.2.
For general k , an exact evaluation for the queue-length distribution is known intwo-queue exhaustive/ k -limited systems with zero setup times (see Lee [11] and Ozawa [15, 16])and with state-dependent switch-over times (see [28]). Remark 3.3.
As discussed in various polling studies, one can readily derive the waiting-time LSTat Q i from F i ( z , z ). Furthermore, there exists a simple relation between the mean waiting times E [ W i ] (at Q i ), i = 1 , , . . . , N in polling systems, a so-called pseudo-conservation law (cf. [4]). Inthe G/1-L case, this pseudo-conservation law reduces to: ρ E [ W ] + ρ (cid:18) − λ E [ S ]1 − ρ (cid:19) E [ W ] = ρ X i =1 λ i E [ B i ]2(1 − ρ ) + ρ E [ S ]2 E [ S ] + E [ S ]2(1 − ρ ) [ ρ + ρ + ρ ] , (6) Throughout the polling literature, G and E seem to have comparable complexity. Their role inthe branching-type polling models, and in (pseudo-)conservation laws, is similar. In view of this,3nd of the simplicity of E/1-L, it is remarkable that G/1-L has remained unsolved for the pasttwenty years, despite the fact that it is a quite relevant model (cf. Bisdikian [1], who introducesa variant of G/1-L as a model for communication networks with bridge-stations and suggests anapproximative approach). Hence we state
Open problem 1.
Determine the joint queue-length PGF at polling instants in the two-queueG/1-L polling system.
In the G/1-L case, (4) holds with h ( z , z ) = β ( z , z ). Obvious attempts to obtain F ( z , z )from (4) include substituting z = 0 (which yields a derivative), and substituting z = h ( z , z )into the lefthand side of (4). In the latter case, one obtains terms F ( h ( h ( z , z ) , z ) , z ) and F ( h ( h ( z , z ) , ,
0) in the righthand side, and iteration does not seem to lead to a solution.Our feeling is that (4), in combination with obvious analyticity conditions of F ( · , · ) inside theproduct of unit circles, leads to a boundary value problem (cf. [7]), but one with a rather compli-cated shift introduced by the function h ( · , · ). Boundary value problems with a shift have beenstudied in the Riemann-Hilbert framework (cf. [8], Section 17), and even in the setting of pollingsystems (cf. [12], which studies a two-queue polling model with Bernoulli service at both queues),but the present problem seems particularly challenging. The next open problem considers the class of polling systems, with N queues, that allow a multi-type branching process interpretation. We are interested in the behaviour of the polling system(under proper scaling conditions), when the deterministic switch-over times tend to infinity. Thislarge switch-over time problem is relevant from a practical point of view, since systems withlarge switch-over times find a wide variety of applications in manufacturing environments (see[26]). Firstly, Subsection 4.1 summarises known results for switch-over time asymptotics in pollingsystems with Poisson arrivals. Subsequently, we pose a conjecture for the behaviour of systems inwhich the arrival process at each of the queues is a general (renewal) process (see Subsection 4.2).Finally, the rigourous proof of this conjecture is stated as an open problem. Under the assumption of Poisson arrival processes , Winands [27] presents an exact asymptoticanalysis of the waiting-time distribution in branching-type polling systems with deterministic switch-over times when the switch-over times tend to infinity. The results of [27] generalise thosederived in [14, 22, 25] for the special case of exhaustive and gated service. Since the waiting timegrows to infinity in the limiting case, [27] focusses on the asymptotic scaled waiting time W i /S as S → ∞ while keeping the ratios of the switch-over times constant. We introduce Φ i as the“exhaustiveness” of the service discipline in Q i , defined as Φ i = 1 − ∂∂z i h i ( z , . . . , z N ) (cid:12)(cid:12) z =1 ,...,z N =1 .Its interpretation is, that each customer present at the start of a visit to Q i will be replaced by anumber of type i customers with mean 1 − Φ i . If Q i receives exhaustive service, the exhaustivenessis 1; for gated service, it is 1 − ρ i . In case of Poisson arrivals and deterministic switch-over times,the distribution of the asymptotic scaled waiting time is given by W i S d −→ − ρ i − ρ U i , ( S → ∞ ) , (7)where U i is uniformly distributed on [ − Φ i Φ i , i ].The closed-form expression of the scaled delay distribution has an intuitively appealing inter-pretation. That is, in the case of increasing deterministic switch-over times the polling systemconverges to a deterministic cyclic system with continuous deterministic service rates 1 / E [ B i ] and4ontinuous demand rates λ i , i = 1 , , . . . , N , which reveals itself, for example, in the fact that thescaled number of customers at Q i at a polling instant of Q i becomes deterministic in the limit asshown in [27]. This means that in the limit the customers arrive to the system and are served atconstant rates with no statistical fluctuation whatsoever and that the scaled queue lengths canbe seen as continuous quantities. Therefore, the uniform distribution emerging in the limitingtheorems can be explained by the fact that it represents the position of the server in the cycle onarrival of a tagged customer. Remark 4.1.
In [23] it is shown that in heavy traffic (HT), i.e., if the load tends to one, theimpact of higher moments of the switch-over times on the waiting-time distribution vanishes.Consequently, the scaled asymptotic waiting time depends on the marginal switch-over time dis-tributions only through the first moment of the total switch-over time in a cycle. Building uponthis observation, [27] analyses the scaled asymptotic waiting time in branching-type polling sys-tems with generally distributed switch-over times under heavy traffic when the switch-over timestend to infinity. The behaviour of the polling system then becomes deterministic, just like inpolling systems with deterministic switch-over times, which are not necessarily operating in HT.
Until now, we have assumed that the arrival processes are Poisson processes. This assumptionis used in [27] to derive the asymptotics presented in the previous subsection, building upon aresult of [3] which derives a strong relation between the waiting-time distributions in models with and without switch-over times. This relation is established by relating the similarities inthe offspring generating functions of the underlying branching processes and by expressing thedifferences between the underlying immigration functions. These results for polling systems with finite switch-over times are exploited and, subsequently, it is shown that significant simplificationsresult as the switch-over times tend to infinity . Unfortunately, the techniques used throughout[27] rely heavily on the Poisson assumption, and corresponding results for polling systems withgeneral arrival processes are not known. Taking a second look at the intuitive interpretationof the aforementioned results, one would expect that the Poisson assumption is not essentialfor this kind of behaviour. That is, if the scaled number of arrivals during an intervisit periodbecomes deterministic, then the length of the scaled visit period (generated by these arrivals)converges to a constant as well. Since the intervisit period is the sum of individual visit periods andswitch-over times, based on strong law of large numbers arguments the scaled number of arrivalssubsequently indeed tends to become deterministic. This circular intuitive reasoning (ignoringthe interdependence between the visit periods) is independent of the precise characteristics of therenewal arrival process chosen. We now pose this statement as a conjecture (see, also, [27]).
Conjecture 4.2.
A cyclic polling system with general (renewal) arrival processes converges to adeterministic cyclic system when the deterministic switch-over times tend to infinity.
To numerically test this conjecture for general arrival processes, we have performed a couple ofsimulation experiments of exhaustive polling systems with general renewal arrivals. In Table 1, weshow results for a symmetric polling system with 3 queues, where the service times are exponentialwith mean 0 .
25. Interarrival times have mean 1 and the corresponding squared coefficient ofvariation (SCV), c A i , is varied between 0 .
25, 0 .
5, 1 and 2. In order to obtain a distribution forthese interarrival times, we fit a phase-type distribution on the first two moments (cf., e.g., [21]).For the cases where the SCV equals 1, Poisson processes are used for the arrival processes in orderto obtain exact results, and this case is included as benchmark.Table 1 shows c P , the SCV of the scaled number of customers at Q at a polling instant of Q for varying values of the marginal switch-over times S i in a cycle. From Table 1, we clearly seethat the coefficient of variation approaches zero when the switch-over times tend to infinity. Forpolling systems with deterministic switch-over times and Poisson arrivals, it can actually be shownanalytically that the SCV of the number of customers in a queue, at the beginning of a visit to5 yclic c A i = 0 . c A i = 0 . c A i = 1 c A i = 2 S i = 1 0 .
121 0 . . S i = 10 0 .
012 0 . . S i = 100 0 .
001 0 . . Table 1: Squared coefficient of variation of the scaled number of customers at Q at a pollinginstant of Q . Values in italic are not obtained by simulation, but are computed analytically.this queue, is inversely proportional to the total switch-over time S . Table 1 seems to suggest thatthis also holds for other arrival processes. It goes without saying that a highly variable arrivalprocess has a negative impact on how “fast” the limiting behaviour is approached. Via Chebyshev’sinequality (see, e.g., [17]) we know that a random variable with zero variance follows a deterministicdistribution and, therefore, this observation provides empirical evidence for the fact that the scalednumber of customers at Q at a polling instant of Q becomes deterministic. Therefore, it confirmsthe validity of our conjecture that the polling system converges to a deterministic cyclic systemas the switch-over times increase to infinity.We have run tests for asymmetric polling systems as well, as shown in Table 2. The first threecolumns show the input parameters: the SCV of the interarrival time distributions, c A i , theimbalance of the interarrival times, I A , and the imbalance of the service times, I B . The imbalanceis the ratio between the largest and the smallest mean interarrival/service time. The arrival ratesand mean service times are chosen such that the differences λ i − λ i +1 and E [ B i +1 ] − E [ B i ] arekept constant for i = 1 , . . . , N −
1. Furthermore, we have chosen the normalisation constraint P Ni =1 λ i /N = 1, implying that the actual arrival rates and mean service times (for fixed ρ ) followfrom the relation ρ = P Ni =1 λ i E [ B i ]. See [2] for a more elaborate description and some examples ofhow the arrival rates and mean service times can be computed from this definition of imbalance.The last two columns of Table 2 contain the SCVs of the waiting times of customers in Q , c W ,and the SCVs of the numbers of customers at Q at a polling instant, c P , for a cyclic pollingsystem with ρ = 0 .
75 and deterministic switch-over times S i = 100, for i = 1 , ,
3. The SCVs ofthe waiting times approach the limiting value , which is the SCV of a uniform distribution, quiterapidly. Furthermore, c P becomes negligibly small, illustrating that the behaviour of the systembecomes deterministic. Cyclic c A i I A I B c W c P .
25 1 1 0 .
335 0 . .
25 1 3 0 .
335 0 . .
25 3 1 0 .
334 0 . .
25 3 3 0 .
335 0 . .
336 0 . .
337 0 . .
336 0 . .
337 0 . Q at a polling instant of Q . Values in italic are not obtained by simulation, but are computedanalytically. 6ummarising, we state the second open problem for polling systems. Open problem 2.
Provide a rigourous proof of Conjecture 4.2, which states that a cyclic pollingsystem with general (renewal) arrival processes converges to a deterministic cyclic system whenthe deterministic switch-over times tend to infinity.
We wish to end the present paper with stating a related open problem given in [14]. That is, [14]shows via numerical testing that similar limit theorems as presented here carry over to systemswith Poisson arrivals and dynamic visit orders (i.e., there exists no pre-determined order in whichthe queues are served). This phenomenon is intuitively explained via heuristic strong law reasoningin [14] and it is conjectured that the limit theorems hold so long as the switch-overs perform aregulating effect . As an example, we show simulation results in Table 3 for the same symmetricpolling systems as studied in Table 1, but now the server switches to the longest queue at the endof a visit. We can see clearly, that the system becomes deterministic as well. A resulting openproblem is, therefore, the classification of polling systems in terms of service, visit and schedulingdisciplines, which exhibit the discussed behaviour.
Longest queue c A i = 0 . c A i = 0 . c A i = 1 c A i = 2 S i = 1 0.125 0.170 0.254 0.434 S i = 10 0.012 0.017 0.026 0.044 S i = 100 0.001 0.002 0.003 0.004 Table 3: Squared coefficient of variation of the scaled number of customers at Q at a pollinginstant of Q . There is a huge literature on polling systems, due to their great applicability in real-life situations.In this paper we have described two problems that have remained unsolved in the polling literature,despite their practical relevance, and despite the fact that seemingly minor adaptations of theseproblems can be solved explicitly. For the first problem, which is the exact analysis of a two-queue polling system with respectively gated and 1-limited service, we pinpoint the difficultiesone runs into when applying standard techniques. The second problem is the analysis of a pollingsystem with general renewal arrivals under the limiting situation where the (deterministic) switch-over times tend to infinity. For this problem we have posed a strong conjecture stating that the(known) results for Poisson arrivals carry over to the system with general renewal arrivals. Byposing these open problems we hope to provide a motivation to search for alternative ways tostudy and hopefully even solve them.
References [1] C. Bisdikian. A queueing model with applications to bridges and the DQDB (IEEE 802.6)MAN.
Computer Networks and ISDN Systems , 25(12):1279–1289, 1993.[2] M. A. A. Boon, E. M. M. Winands, I. J. B. F. Adan, and A. C. C. van Wijk. Closed-formwaiting time approximations for polling systems.
Performance Evaluation , 68:290–306, 2011.[3] S. C. Borst and O. J. Boxma. Polling models with and without switchover times.
OperationsResearch , 45(4):536 – 543, 1997.[4] O. Boxma. Workload and waiting times in single-server systems with multiple customerclasses.
Queueing Systems , 5:185 – 214, 1989.75] O. J. Boxma and W. P. Groenendijk. Two queues with alternating service and switchingtimes. In O. J. Boxma and R. Syski, editors,
Queueing Theory and its Applications – LiberAmicorum for J.W. Cohen , pages 261–282. North-Holland Publishing Company, Amsterdam,1988.[6] J. W. Cohen and O. J. Boxma. The
M/G/
Proc. Performance ’81 ,pages 181–199. North-Holland Publishing Company, Amsterdam, 1981.[7] J. W. Cohen and O. J. Boxma.
Boundary Value Problems in Queueing System Analysis .North-Holland Publishing Company, Amsterdam, 1983.[8] F. D. Gakhov.
Boundary Value Problems . Pergamon Press, Oxford, 1966.[9] W. P. Groenendijk.
Conservation Laws in Polling Systems . PhD thesis, University of Utrecht,1990.[10] O. C. Ibe. Analysis of polling systems with mixed service disciplines.
Stochastic Models , 6(4):667–689, 1990.[11] D.-S. Lee. A two-queue model with exhaustive and limited service disciplines.
StochasticModels , 12(2):285–305, 1996.[12] D.-S. Lee. Analysis of a two-queue model with Bernoulli schedules.
Journal of AppliedProbability , 34:176–191, 1997.[13] H. Levy and M. Sidi. Polling systems: applications, modeling, and optimization.
IEEETransactions on Communications , 38:1750–1760, 1990.[14] T. Olsen. Limit theorems for polling models with increasing setups.
Probability in the Engi-neering and Informational Sciences , 15:35–55, 2001.[15] T. Ozawa. Alternating service queues with mixed exhaustive and K -limited services. Perfor-mance Evaluation , 11(3):165–175, 1990.[16] T. Ozawa. Waiting time distributions in a two-queue model with mixed exhaustive and gated-type K -limited services. In Proceedings of International Conference on the Performance andManagement of Complex Communication Networks , pages 231–250, Tsukuba, 1997.[17] A. Papoulis.
Probability, Random Variables, and Stochastic Processes . McGraw-Hill, NewYork, 2 nd edition, 1984.[18] J. A. C. Resing. Polling systems and multitype branching processes. Queueing Systems , 13:409 – 426, 1993.[19] L. Tak´acs. Two queues attended by a single server.
Operations Research , 16(3):639–650, 1968.[20] H. Takagi. Queuing analysis of polling models.
ACM Computing Surveys (CSUR) , 20:5–28,1988.[21] H. C. Tijms.
Stochastic Models: An Algorithmic Approach . Wiley, Chichester, 1994.[22] R. D. van der Mei. Delay in polling systems with large switch-over times.
Journal of AppliedProbability , 36:232–243, 1999.[23] R. D. van der Mei. Towards a unifying theory on branching-type polling models in heavytraffic.
Queueing Systems , 57:29–46, 2008.[24] V. M. Vishnevskii and O. V. Semenova. Mathematical methods to study the polling systems.
Automation and Remote Control , 67(2):173–220, 2006.825] E. M. M. Winands. On polling systems with large setups.
Operations Research Letters , 35:584–590, 2007.[26] E. M. M. Winands.
Polling, Production & Priorities . PhD thesis, Eindhoven University ofTechnology, 2007.[27] E. M. M. Winands. Branching-type polling systems with large setups.