Expected waiting times in polling systems under priority disciplines

In this paper, we analyse a service system which consists of several queues (stations) polled by a single server in a cyclic order with arbitrary switchover times. Customers from several priority classes arrive into each of the queues according to independent Poisson processes and require arbitrarily distributed service times. We consider the system under various priority service disciplines: head-of-the-line priority limited to one and semi-exhaustive, head-of-the-line priority limited to one with background customers, and global priority limited to one. For the first two disciplines we derive a pseudo conservation law. For the third discipline, we show how to obtain the expected waiting time of a customer from any given priority class. For the last discipline we find the expected waiting time of a customer from the highest priority class. The principal tool for our analysis is the stochastic decomposition law for a single server system with vacations.     

Approximating queue size and waiting time distributions in general polling systems

Polling system models are extensively used to model a large variety of computer and communication networks as well as production and service systems in which multiple customer classes or a number of distinct items compete for the capacity of a common server or production facility. In this paper we describe an efficient approximation method for the steady state distributions of the queue sizes and waiting times. This method is highly accurate as demonstrated by an extensive numerical study. In addition, it is highly adaptable to a variety of arrival patterns and switching protocols, including exhaustive and gated regimes, simple cyclical systems as well as general polling tables. For a system withN stations, one finds the firstK probability density function values of the steady state queue size in any given station inO(max(N, K ²) time only. When executed on an IBM system RS/6000, we have observed an average CPU time of less than 1 second for systems with as many as 50 stations over a large variety of parameter settings.     

Stability of non-Markovian polling systems

A stationary regime for polling systems with general ergodic (G/G) arrival processes at each station is constructed. Mutual independence of the arrival processes is not required. It is shown that the stationary workload so constructed is minimal in the stochastic ordering sense. In the model considered the server switches from station to station in a Markovian fashion, and a specific service policy is applied to each queue. Our hypotheses cover the purely gated, thea-limited, the binomial-gated and other policies. As a by-product we obtain sufficient conditions for the stationary regime of aG/G/1/ queue with multiple server vacations (see Doshi [11]) to be ergodic.Work presented at the INRIA/ORSA Conference on Applied Probability in Engineering, Computer and Communication Sciences, Paris, June 16–18, 1993.     

A note on polling models with renewal arrivals and nonzero switch-over times

We consider gated polling systems with general service and switch-over times and with renewal arrival processes. We derive closed-form expressions for the expected delay in heavy-traffic (HT). So far, proofs of HT limits have only been obtained for Poisson-type arrival processes, whereas for renewal arrivals limits are based on conjectures.     

Mean value analysis for polling systems

The present paper deals with the problem of calculating mean delays in polling systems with either exhaustive or gated service. We develop a mean value analysis (MVA) to compute these delay figures. The merits of MVA are in its intrinsic simplicity and its intuitively appealing derivation. As a consequence, MVA may be applied, both in an exact and approximate manner, to a large variety of models.     

Towards a unifying theory on branching-type polling systems in heavy traffic

For a broad class of polling models the evolution of the system at specific embedded polling instants is known to constitute a multi-type branching process (MTBP) with immigration. In this paper it is shown that for this class of polling models the vector that describes the state of the system at these polling instants, say X=(X ₁,…,X _M ), satisfies the following heavy-traffic behavior (under mild assumptions):

Polling systems with state-dependent setup times

We analyze a polling system with multiple stations (queues) attended by a cycling server, in which a setup occurs only when the queue that is polled by the server has one or more customers present. Although such systems are appropriate for modeling numerous manufacturing and telecommunication systems, their analysis is not well developed in the literature. We provide an exact analysis for the 2 station model and present two approximation schemes to determine the mean station waiting times for models with 3 or more stations. We show that some approximate models which have been proposed in the literature for providing upper bounds on the mean station waiting times do not always yield upper bounds. Extensive numerical tests indicate that a simple average of the two approximation schemes yields a close estimate of the true mean station waiting time. This average-of-approximations procedure appears to be robust for a large range of parameter values.Research supported in part by the Natural Sciences and Engineering Research Council of Canada under grant OGP0045904.Research supported in part by the National Science Foundation under grant DMI-9500471.     

On polling systems with large setups

Polling systems with large deterministic setup times find many applications in production environments. We study the delay distribution in exhaustive polling systems when the setup times tend to infinity. Via mean value analysis a novel approach is developed to show that the scaled delay distribution converges to a uniform distribution.     

Stability and continuity of polling systems

The stability of a polling system with exhaustive service and a finite number of users, each with infinite buffers is considered. The arrival process is more general than a Poisson process and the system is not slotted. Stochastic continuity of the stationary distributions, rates of convergence and functional limit theorems for the queue length and waiting time processes have also been proved. The results extend to the gated service discipline.     

Queue lengths and workloads in polling systems

For a single-server multi-station polling system, we focus on the generating function and Laplace–Stieltjes transform of the time-stationary joint queue length and workload distributions, respectively, under no further assumptions on the service discipline. We express these quantities as expressions involving the generating functions of the joint queue length distribution at visit beginnings and visit completions at the various stations. The latter is known for a broad variety of cases. Finally, we identify a workload decomposition result.     

Relating polling models with zero and nonzero switchover times

We consider a system ofN queues served by a single server in cyclic order. Each queue has its own distinct Poisson arrival stream and its own distinct general service-time distribution (asymmetric queues), and each queue has its own distinct distribution of switchover time (the time required for the server to travel from that queue to the next). We consider two versions of this classical polling model: In the first, which we refer to as the zero-switchover-times model, it is assumed that all switchover times are zero and the server stops traveling whenever the system becomes empty. In the second, which we refer to as the nonzero-switchover-times model, it is assumed that the sum of all switchover times in a cycle is nonzero and the server does not stop traveling when the system is empty. After providing a new analysis for the zero-switchover-times model, we obtain, for a host of service disciplines, transform results that completely characterize the relationship between the waiting times in these two, operationally-different, polling models. These results can be used to derive simple relations that express (all) waiting-time moments in the nonzero-switchover-times model in terms of those in the zero-switchover-times model. Our results, therefore, generalize corresponding results for the expected waiting times obtained recently by Fuhrmann [Queueing Systems 11 (1992) 109—120] and Cooper, Niu, and Srinivasan [to appear in Oper. Res.].Research supported in part by the National Science Foundation under grant DDM-9001751.     

Markovian polling systems with mixed service disciplines and retrial customers

A Markovian polling model with a mixture of exhaustive and gated type stations is considered. The cuttomers are ofn different tppes and arrive to the system acccording to the Poisson distribution, in batches containing customers of all types (correlated batch arrivals). The customers who find upon arrival the server unavailable repeat their arrival individually after a random amount of time (retrial customers). The service timesT _i and the switchover timesV _ij are arbitrarily distributed with different distributions for the different stations. For such a model we obtain formulae for the expected number of customers in each station in a steady state. Our formulae hold also for zero switchover periods and can easily be adapted to hold for the corresponding ordinary Markovian mixed polling models with/without switchover times and correlated batch arrivals. Numerical calculations are finally used to observe system's performance.     

The waiting time distribution for a correlated queue with exponential interarrival and service times

We are concerned with the analysis of the waiting time distribution in an $M∕M∕1$ queue in which the interarrival time between the $n$th and the $(n+1)$th customers and the service time of the $n$th customer are correlated random variables with Downton’s bivariate exponential distribution. In this paper we show that the conditional waiting time distribution, given that the waiting time is positive, is exponential.     

Waiting times in a two-queue model with exhaustive and Bernoulli service

This paper deals with waiting times in a two-queue polling system in which one queue is served according to the Bernoulli service discipline and the other one attains exhaustive service. Exact results are derived for the LST's of the waiting time distributions via an iteration scheme. Based on those results the mean waiting times are expressed in the system parameters.     

Stability,monotonicity and invariant quantities in general polling systems

Consider a polling system withK1 queues and a single server that visits the queues in a cyclic order. The polling discipline in each queue is of general gated-type or exhaustive-type. We assume that in each queue the arrival times form a Poisson process, and that the service times, the walking times, as well as the set-up times form sequences of independent and identically distributed random variables. For such a system, we provide a sufficient condition under which the vector of queue lengths is stable. We treat several criteria for stability: the ergodicity of the process, the geometric ergodicity, and the geometric rate of convergence of the first moment. The ergodicity implies the weak convergence of station times, intervisit times and cycle times. Next, we show that the queue lengths, station times, intervisit times and cycle times are stochastically increasing in arrival rates, in service times, in walking times and in setup times. The stability conditions and the stochastic monotonicity results are extended to the polling systems with additional customer routing between the queues, as well as bulk and correlated arrivals. Finally, we prove that the mean cycle time, the mean intervisit time and the mean station times are invariant under general service disciplines and general stationary arrival and service processes.     

Exact analysis of asymmetric random polling systems with single buffers and correlated input process

We introduce a simple approach for modeling and analyzing asymmetric random polling systems with single buffers and correlated input process. We consider two variations of single buffers system: the conventional system and the buffer relaxation system. In the conventional system, at most one customer may be resided in any queue at any time. In the buffer relaxation system, a buffer becomes available to new customers as soon as the current customer is being served. Previous studies concentrate on conventional single buffer system with independent Poisson process input process. It has been shown that the asymmetric system requires the solution ofm 2 ^m –1) linear equations; and the symmetric system requires the solution of 2 ^m–1–1 linear equations, wherem is the number of stations in the system. For both the conventional system and the buffer relaxation system, we give the exact solution to the more general case and show that our analysis requires the solution of 2 ^m –1 linear equations. For the symmetric case, we obtain explicit expressions for several performance measures of the system. These performance measures include the mean and second moment of the cycle time, loss probability, throughput, and the expected delay observed by a customer.     

Dominance relations in polling systems

In this paper we compare several service disciplines commonly used in polling systems. We present a sample path comparison which allows us to evaluate the efficiency of the different policies based on thetotal amount of work found in the systemat any time. The analysis is carried out for a large variety of polling schemes under fairly general conditions and can be used to construct a hierarchy of the different service schemes.