Optimal Control of Parallel Queues with Batch Service (original) (raw)
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Numerical Analysis of Optimal Control Policies for Queueing Systems with Heterogeneous Servers
INTRODUCTION The problem of optimal jobs assignment to heterogeneous servers arises in many applications. The problem of optimal jobs assignment for two heterogeneous servers with respect to minimization of long run average mean number jobs in the system was considered in [1], where it was shown that the policy, which minimizes the number of customers in the system, has a threshold property and consists in using the fastest server if necessary. For the multi-server system, these properties of an optimal policy were generalized in [2]. In the talk, an algorithm is proposed which gives the possibility to find optimal threshold levels for di#erent values of system parameters and investigate their behavior. Some numerical examples are also included. 2. THE PROBLEM Consider an M/M/K/N -K (K N<#) controllable queuing system with K heterogeneous exponential servers of intensities k (k = 1,K),N-K places in the bu#er, and a Poisson input of jobs with the intensity #. At the arrival ti
Threshold Policy for M / M / C / K Queue with Integrated Traffic
2013
In this paper, we discuss a finite capacity queue with integrated traffic and batch services under N-policy. There are two types of customer arrive according to poisson distribution. As soon as the queue size reaches the threshold level N, the all server C are turned on and serves both types of traffic one by one upto threshold level d( >C) of the customers in the system. After the threshold level d, the server provides the service of the type 1 customers in a batch whereas type 2 customers are lost. The queue size distribution is derived with the help of recursive method. Various performance measures, i.e. expected number of customers in the queue and in the system, probability of the server being turn off, under setup, busy and the expected idle/busy period etc. are determined.
Effect of the server capacity distribution on the optimal control of a bulk service queueing system
Chaos, Solitons & Fractals, 2003
ABSTRACT In this paper the optimal control of a single-channel bulk service queueing system with random server capacity is investigated. Given an accumulation level r, the server stops processing new customers whenever the queue falls below r and resumes service when the queue reaches level r again. Server capacity becomes random following an idle period. A quick search procedure is designed to determine the value of r that yields the minimum expected total cost per unit of time. The effect of the server capacity distribution on the optimal control is then studied. Finally, a sensitivity analysis is conducted in order to assess the extent to which our results are valid.
Mathematics
In this paper, we study the problem of optimal routing for the pair of two-server heterogeneous queues operating in parallel and subsequent optimal allocation of customers between the servers in each queue. Heterogeneity implies different servers in terms of speed of service. An open-loop control assumes the static resource allocation when a router has no information about the state of the system. We discuss here the algorithm to calculate the optimal routing policy based on specially constructed Markov-modulated Poisson processes. As an alternative static policy, we consider an optimal Bernoulli splitting which prescribes the optimal allocation probabilities. Then, we show that the optimal allocation policy between the servers within each queue is of threshold type with threshold levels depending on the queue length and phase of an arrival process. This dependence can be neglected by using a heuristic threshold policy. A number of illustrative examples show interesting properties o...
Queueing analysis and optimal control of and systems
Computers & Industrial Engineering, 2009
We first consider a finite-buffer single server queue where arrivals occur according to batch Markovian arrival process ðBMAPÞ: The server serves customers in batches of maximum size 'b' with a minimum threshold size 'a'. The service time of each batch follows general distribution independent of each other as well as the arrival process. We obtain queue length distributions at various epochs such as, pre-arrival, arbitrary, departure, etc. Some important performance measures, like mean queue length, mean waiting time, probability of blocking, etc. have been obtained. Total expected cost function per unit time is also derived to determine the optimal value N Ã of N at a minimum cost for given values of a and b. Secondly, we consider a finite-buffer single server queue where arrivals occur according to BMAP and service process in this case follows a non-renewal one, namely, Markovian service process ðMSPÞ: Server serves customers according to general bulk service rule as described above. We derive queue length distributions and important performance measures as above. Such queueing systems find applications in the performance analysis of communication, manufacturing and transportation systems.
Optimal queueing systems controls with finite buffers and with multiple component cost functions
IEEE Transactions on Systems, Man, and Cybernetics
Ahstruct-Analytical models of stochastic discrete event dynamic systems are now widely used to control congestion while maintaining throughput in many data communication, computing and production networks. An analytical and tractable framework of an optimal feedback control policy is developed here for queueing networks with limited storage buffers and with sequence-dependent service times. The general response measure used incorporates delay, shortage, and the setup costs. This formulation provides a new perspective to dynamic flow control models that are based on a rational function of the queue lengths of various downstream queues rather than on time. This formulation also enables the derivation of various performance measures for each node. Presented measures include throughput rate, utilization, queue length, starvation intervals and setups. Detailed numerical examples provide some insights into the structure and performance of the optimal policy. It is explicated that the buffer capacity vector has a minimal effect on the routing decisions in the interior of the state space and that the relative value of the buffer spaces declines with the increase in the delay costs. The impact of major resource allocation decisions and of changes in the performance cost measures are also explained.
Advances in Operations Research, 2015
We analyze an infinite-buffer batch-size-dependent batch-service queue with Poisson arrival and arbitrarily distributed service time. Using supplementary variable technique, we derive a bivariate probability generating function from which the joint distribution of queue and server content at departure epoch of a batch is extracted and presented in terms of roots of the characteristic equation. We also obtain the joint distribution of queue and server content at arbitrary epoch. Finally, the utility of analytical results is demonstrated by the inclusion of some numerical examples which also includes the investigation of multiple zeros.
Constrained Load-Balancing Policies for Parallel Single-Server Queue Systems
Management Science, 2020
Flow-control policies that balance server loads are well known for improving performance of queueing systems with multiple nodes. However, although load balancing benefits the system overall, it may negatively impact some of the queueing nodes. For example, it may reduce throughput rates or engender unfairness with respect to some performance measures. For queueing systems with multiple single-server nodes, we propose a set of constrained load-balancing policies that ensures the expected arrival rate to each queueing node is not reduced, and we show that such policies provide multiple benefits for each queueing node: stochastically fewer customers and lower variance of the number of customers at each queueing node. These results imply performance improvement as measured by multiple general objective functions, including but not limited to the expected number of customers at a queueing node, probability of having a high number of customers, variance of the number of customers, and ex...
Performance Evaluation, 2011
Most research concerning batch-service queueing systems has focussed on some specific aspect of the buffer content. Further, the customer delay has only been examined in the case of single arrivals. In this paper, we examine three facets of a threshold-based batch-service system with batch arrivals and general service times. First, we compute a fundamental formula from which an entire gamut of known as well as new results regarding the buffer content of batch-service queues can be extracted. Secondly, we produce accurate light-and heavy-traffic approximations for the buffer content. Thirdly, we calculate various quantities with regard to the customer delay. This paper thus provides a whole spectrum of tools to evaluate the performance of batch-service systems.