Switchover Time Research Articles

Polling Systems in Heavy Traffic: A Bessel Process Limit

This paper studies the classical polling model under the exhaustive-service assumption; such models continue to be very useful in performance studies of computer/communication systems. The analysis here extends earlier work of the authors to the general case of nonzero switchover times. It shows that, under the standard heavy-traffic scaling, the total unfinished work in the system tends to a Bessel-type diffusion in the heavy-traffic limit. It verifies in addition that, with this change in the limiting unfinished-work process, the averaging principle established earlier by the authors carries over to the general model.

Analysis of demand-priority access using a priority queue with server vacations and message dependent switchover times

In this paper, we analyse the message waiting times in a local area network (LAN) that uses the demand?priority access method. This is a priority?based round?robin arbitration method where the central controller (the repeater) polls its connected ports to determine which have transmission requests pending and the requests' priority classes. We model it as a 2?priority M/G/1 queue with multiple server vacations and switchover time between service periods. The service discipline is nonpreemptive and the length of the switchover time is dependent upon the priority class of the preceding message served as well as that of the message to be served next. We provide an approximate analysis for the waiting times of both message classes and derive expressions for the Laplace---Stieltjes transforms (LST) of the stationary distributions and the mean waiting times. We conclude with numerical and simulation results to show the applicability and accuracy of the analytical model.

Threshold start-up control policy for polling systems

A threshold start-up policy is appealing for manufacturing (service) facilities that incur a cost for keeping the machine (server) on, as well as for each restart of the server from its dormant state. Analysis of single product (customer) systems operating under such a policy, also known as the N-policy, has been available for some time. This article develops mathematical analysis for multiproduct systems operating under a cyclic exhaustive or globally gated service regime and a threshold start-up rule. It pays particular attention to modeling switchover (setup) times. The analysis extends/unifies existing literature on polling models by obtaining as special cases, the continuously roving server and patient server polling models on the one hand, and the standard M/G/1 queue with N-policy, on the other hand. We provide a computationally efficient algorithm for finding aggregate performance measures, such as the mean waiting time for each customer type and the mean unfinished work in system. We show that the search for the optimal threshold level can be restricted to a finite set of possibilities.

Dynamic server assignment in a two-queue model

We consider a polling model of two M/G/1 queues, served by a single server. The service policy for this polling model is of threshold type. Service at queue 1 is exhaustive. Service at queue 2 is exhaustive unless the size of queue 1 reaches some level T during a service at queue 2; in the latter case the server switches to queue 1 at the end of that service. Both zero- and nonzero switchover times are considered. We derive exact expressions for the joint queue length distribution at customer departure epochs, and for the steady-state queue-length and sojourn time distributions. In addition, we supply a simple and very accurate approximation for the mean queue lengths, which is suitable for optimization purposes.

Nonlinear time-independent and time-dependent propagation of light in direct-gap semiconductors with paired excitons bound into biexcitons

We study a new class of nonlinear cooperative phenomena that occur when light propagates in direct-gap semiconductors. The nonlinearity here is due to a process, first discussed by A. L. Ivanov, L. V. Keldysh, and V. V. Panashchenko, in which two excitons are bound into a biexciton by virtue of their Coulomb interaction. For the geometry of a ring cavity, we derive a system of nonlinear differential equations describing the dynamical evolution of coherent excitons, photons, and biexcitons. For the time-independent case we arrive at the equation of state of optical bistability theory, and this equation is found to differ considerably from the equations of state in the two-level atom model and in the exciton region of the spectrum. We examine the stability of the steady states and determine the switchover times between the optical bistability branches. We also show that in the unstable sections of the equation of state, nonlinear periodic and chaotic self-pulsations may arise, with limit cycles and strange attractors being created in the phase space of the system. The scenario for the transition to the dynamical chaos mode is found. A computer experiment is used to study the dynamic optical bistability. Finally, we discuss the possibility of detecting these phenomena in experiments.

Polling Models With and Without Switchover Times

We consider two different single-server cyclic polling models: (i) a model with zero switchover times, and (ii) a model with nonzero switchover times, in which the server keeps cycling when the system is empty. For both models we relate the steady-state queue length distribution at a queue to the queue length distributions at server visit beginning and visit completion instants at that queue; as a by-product we obtain a short proof of the Fuhrmann-Cooper decomposition. For the large class of polling systems that allow a multitype branching process interpretation, we expose a strong relation between the queue length, as well as waiting-time, distributions in the two models. The results enable a very efficient numerical computation of the waiting-time moments under different switchover time scenarios.

A Closed Form Solution for the Asymmetric Random Polling System with Correlated Levy Input Process

We introduce a simple approach for modeling and analyzing the random polling system with asymmetric arrival rates, service times, and switchover times. It is assumed that the customer arrival processes at all queues are correlated Levy input processes. Two classes of service disciplines, random gated and 1-limited, are considered. The random gated service discipline generalizes several known service disciplines. We obtain explicit expressions for several performance measures of the system. These performance measures include the mean and second moment of the cycle time, the queue length at the beginning of a cycle of service and the expected delay observed by a customer. For the special case of independent Poisson input processes at all queues, we also provide new proof of several well-known pseudo-conservation laws.

Optimal Control for an Mx/G/1 Queue with Two Operation Modes

The controlled Mx/G/1-type queueing model with two modes of operation is considered. The modes are characterized by different service time distributions and input rates. The switchover times are imposed in the model. The embedded stationary queue-length distribution and the explicit dependence of operation criteria on switchover levels are derived.

A POLLING MODEL WITH RETRIAL CUSTOMERS

A polling model with n stations and switchover times is considered. The customers are of n different types, arrive to the system according to the Poisson distribution in batches of random size, and if they find the server unavailable, they start to make retrials until succeed to find a position for service. Each batch may contain customers of different types while the numbers of customers belonging to each type in a batch are distributed according to a multivariate general distribution. The server, upon polling a station, stays there for an exponential period of time and if a customer asks for service before this time expires, the customer is served and a new "stay period" begins. Finally the service times and the switchover times are both arbitrarily distributed with different distributions for the different stations. For such a model we obtain formulas for the expected number of retrial customers in each station in a steady state. Our results can be easily adapted to hold for zero switchover times and also in the case of the ordinary exhaustive service polling model with (without) switchover times and correlated batch arrivals. In all cases mentioned above (retrial model, exhaustive model, switchover times, zero switchover times) to find the expected queue lengths we need finally to solve a set of only n linear equations (O(n^3) arithmetic operations to compute the coefficients). Tables of numerical values are finally obtained and used to observe the system performance when we vary the values of the parameters.

Performance evaluation of PC routers using a single-server multi-queue system with a reflection technique

In this paper, a cyclic service system with switch-over times introduced by K.S. Watson is used to define the characteristics of the queueing model of a PC router developed at La Trobe University, and based on Vance Morrison's PC router. Two methods of approximating its performance are investigated. The first one is to modify the mean waiting-time approximations for a single-server multi-queue system with nonexhaustive cyclic service, and with non-zero switch-over times of the server between consecutive queues introduced by O.J. Boxma and B. Meister, while the second is to develop a tool based on a reflection technique. The second method is chosen to measure the delay for three packet sizes (64, 576 and 1514 bytes) with four types of PC router (our PC router with and without IP checksum verification, and Vance Morrison's PC router—packet driver and non-packet driver versions). The results of performance analysis are presented.

Message delay for a priority-based automatic meter reading network

In this paper, we model a data acquisition system as cyclic polling systems with two priority levels. Two types of polling system are studied, exhaustive and time limited service, with application to automatic meter reading. In the case of the exhaustive service strategy, stations are served exhaustively for both priorities, with high priority messages served first on a non-preemptive basis. For the case of the time limited discipline, each station is served for no more than τ max but not less than τ min. Approximate results are obtained for the high and low priority mean waiting time for polling systems with constant switchover times, N stations, general service times distributions, and independent arrival processes for each priority level. Comparisons of the approximations with simulation results for different service distributions, traffic intensities and load combinations show a high degree of accuracy.

A two-stage tandem queue attended by a moving server with holding and switching costs

We consider a two-stage tandem queue attended by a moving server, with homogeneous Poisson arrivals and general service times. Two different holding costs for stages 1 and 2 and different switching costs from one stage to the other are considered. We show that the optimal policy in the second stage is greedys and if the holding cost rate in the second stage is greater or equal to the rate in the first stage, then the optimal policy in the second stage is also exhaustive. Then, the optimality condition for sequential service policy in systems with zero switchover times is introduced. Considering some properties of the optimal policy, we then define a Triple-Threshold (TT) policy to approximate the optimal policy in the first stage. Finally, a model is introduced to find the optimal TT policy, and using numerical results, it is shown that the TT policy accurately approximates the optimal policy.

A Decomposition Theorem for Polling Models: The Switchover Times are Effectively Additive

We consider the classical polling model: queues served in cyclic order with either exhaustive or gated service, each with its own distinct Poisson arrival stream, service-time distribution, and switchover-time (the server's travel time from that queue to the next) distribution. Traditionally, models with zero switchover times (the server travels at infinite speed) and nonzero switchover times have been considered separately because of technical difficulties reflecting the fact that in the latter case the mean cycle time approaches zero as the travel speed approaches infinity. We argue that the zero-switchover-times model is the more fundamental model: the mean waiting times in the nonzero-switchover-times model decompose (reminiscent of vacation models) into a sum of two terms, one being a simple function of the sum of the mean switchover times, and the other the mean waiting time in a “corresponding” model obtained from the original by setting the switchover times to zero and modifying the service-time variances. This generalizes a recent result of S. W. Fuhrmann for the case of constant switchover times, where no variance modification is necessary. The effect of these studies is to reduce computation and to improve theoretical understanding of polling models.

Proof of stability conditions for token passing rings by Lyapunov functions

A token passing ring can be described as a system of M queues with one server that rotates around the queues sequentially. Georgiadis-Szpankowski (1992) considered rings where the token (server) performs x /spl nabla/ l/sub j/ services on queue j, where x is the size of queue j upon arrival of the token, and l/sub j/ is a fixed limit of service for queue j. The token then spends some random time switching to the next queue. For j=1, ..., M, arrivals to queue j are Poisson with rate /spl lambda//sub j/, and service times have mean s/sub j/ and are independent of the arrival and switchover processes. The purpose of this paper is to give an alternate and simpler proof of the stability conditions given by Georgiadis-Szpankowski using Lyapunov functions. An additional assumption is made about the second moments of the service and switchover times being finite.

Computing distributions and moments in polling models by numerical transform inversion

We show that probability distributions and moments of performance measures in many polling models can be effectively computed by numerically inverting transforms (generating functions and Laplace transforms). We develop new efficient iterative algorithms for computing the transform values and then use our recently developed variant of the Fourier-series method to perform the inversion. We also show how to use this approach to compute moments and asymptotic parameters of the distributions. We compute a two-term asymptotic expansion of the tail probabilities, which turns out to be remarkably accurate for small tail probabilities. The tail probabilities are especially helpful in understanding the performance of different polling disciplines. For instance, it is known that the exhaustive discipline produces smaller mean steady-state waiting times than the gated discipline, but we show that the reverse tends to be true for small tail probabilities. The algorithms apply to describe the transient behavior of stationary or non-stationary models as well as the steady-state behavior of stationary models. We demonstrate effectiveness by analyzing the computational complexity and by doing several numerical examples for the gated and exhaustive service disciplines, with both zero and non-zero switchover times. We also show that our approach applies to other polling models. Our main focus is on computing exact tail probabilities and asymptotic approximations to them, which seems not to have been done before. However, even for mean waiting times, our algorithm is faster than previous algorithms for large models. The computational complexity of our algorithm is O( N α ) for computing performance measures at one queue and O( N 1+ α ) for computing performance measures at all queues, where N is the number of queues and α is typically between 0.6 and 0.8.

When Should a Roving Server Be Patient?

When polling systems are used to model real-world systems, it is typically assumed that the server switches continuously (“roves”) even when there are no waiting jobs in the system. However, requiring the server to be patient, instead of having it rove, might be more realistic. Furthermore, operational control of these systems can be improved by knowing answers to questions like “under what circumstances should be roving server be patient?” and “at which stations?”. This paper analyzes the patient server model and provides explicit expressions for the waiting time distributions, the mean waiting times and the pseudo-conservation law. Several variants of the patient server model are considered. We show that while the patient server mechanism is generally better than the roving server mechanism in the work-in-process (WIP) reduction sense, there do exist cases where roving is better. Counter-intuitive examples where reducing switchover time can increase WIP are also reported.

Sojourn time analysis for a cyclic-service tandem queueing model with general decrementing service

A sojourn time analysis is provided for a cyclic-service tandem queue with general decrementing service which operates as follows: starting once a service of queue 1 in the first stage, a single server continues serving messages in queue 1 until either queue 1 becomes empty, or the number of messages decreases to k less than that found upon the server's last arrival at queue 1, whichever occurs first, where 1 ≤ k ≤ ∞. After service completion in queue 1, the server switches over to queue 2 in the second stage and serves all messages in queue 2 until it becomes empty. It is assumed that an arrival stream is Poissonian, message service times at each stage are generally distributed and switch-over times are zero. This paper analyzes joint queue-length distributions and message sojourn time distributions.

Polling Systems with Zero Switchover Times: A Heavy-Traffic Averaging Principle

In polling systems, $M \geq 2$ queues are visited by a single server in cyclic order. These systems model such diverse applications as token-ring communication networks and cyclic production systems. We study polling systems with exhaustive service and zero switchover (walk) times. Under standard heavy-traffic assumptions and scalings, the total unfinished work converges to a one-dimensional reflected Brownian motion, whereas the workloads of individual queues change at a rate that becomes infinite in the limit. Although it is impossible to obtain a multidimensional limit process in the usual sense, we obtain an "averaging principle" for the individual workloads. To illustrate the use of this principle, we calculate a heavy-traffic estimate of waiting times.

Pseudo-conservation law for a priority polling system with mixed service strategies

In this paper we consider a discrete-time cyclic-service system consisting of multiple stations visited by a single server. Several priority classes of customers arrive at each station according to independent batch Bernoulli processes. The head-of-the-line (HL) priority rule and non-zero switch-over times between stations are assumed. The customers are served at each station under a mixed (exhaustive, gated, and 1-limited) service strategy. We obtain an exact expression for a weighted sum of the mean waiting times for the individual priority classes, a so-called pseudo-conservation law. A continuous-time result is derived as a special case of our model.

Optimal Batch Sizing and Repair Strategies for Operations with Repairable Jobs

This paper presents a model of a bottleneck facility that performs two distinct types of operations: “regular” and “repair.” Both switch-over time and cost are incurred when the facility switches from performing one type of operation to a different type. Upon the completion of a batch of jobs in the regular mode, each batch is subjected to a test, where the entire batch (of jobs) will be classified accordingly as either nondefective, repairable, or nonrepairable. A nondefective batch continues its process downstream, a nonrepairable batch is scrapped, and a repairable batch can be cycled back to the bottleneck facility for repair. The objective of this paper is to determine the optimal repair policy for the bottleneck facility so that the long run average operating profit is maximized. We first characterize the optimal repair policy by showing that the optimal repair policy must take one of the two forms: a “repair-none” policy under which all repairable batches are scrapped, or a “repair-all” policy under which all repairable batches are repaired. We then develop optimality conditions for the repair-none policy and the repair-all policy. When the repair-all policy is optimal, we further show that there exists an optimal “threshold” operating policy that can be described as follows: upon completion of a regular batch, switch over to the repair mode only if the number of available repairable batches exceeds a certain threshold value. We also evaluate the impact of batch sizes, yield, and switch-over cost on the optimal operating policy.