Optimal Control of a Two-Server Heterogeneous Queueing System with Breakdowns and Constant Retrials (original) (raw)

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

Approximations in performance analysis of a controllable queueing system with heterogeneous servers

Mathematics, 2020

The paper studies a controllable multi-server heterogeneous queueing system where servers 1 operate at different service rates without preemption, i.e. the service times are uninterrupted. The 2 optimal control policy allocates the customers between the servers in such a way that the mean 3 number of customers in the system reaches its minimal value. The Markov decision model and the 4 policy-iteration algorithm are used to calculate the optimal allocation policy and corresponding mean 5 performance characteristics. The optimal policy, when neglecting the weak influence of slow servers, 6 is of threshold type defined as a sequence of threshold levels which specifies the queue lengths 7 for the usage of any slower server. To avoid time-consuming calculations for systems with a large 8 number of servers, we focus here on a heuristic evaluation of the optimal thresholds and compare this 9 solution with the real values. We develop also the simple lower and upper bound methods based on 10 approximation by an equivalent heterogeneous queueing system with a preemption to measure the 11 mean number of customers in the system operating under the optimal policy. Finally, the simulation 12 technique is used to provide sensitivity analysis of the heuristic solution to changes in the form of 13 inter-arrival and service time distributions. 14 Keywords: Heterogeneous servers; Markov decision process; policy-iteration algorithm; mean 15 number of customers; decomposable semi-regenerative process 16

Optimal Control of Queueing Systems with Heterogeneous Servers

Queueing Systems, 2004

An optimal policy to minimize the queue length in a multi-server controllable queueing system with heterogeneous servers has a threshold property, and it uses the fastest server if necessary (see and ). This study gives a numerical description of optimal policies that minimize the operational cost for such a system.

Threshold control policies for heterogeneous server systems

Mathematical Methods of Operations Research (ZOR), 2002

We study the problem of optimally controlling a multiserver queueing system. Customers arrive in a Poisson fashion and join a single queue, served by N servers, S 1 ; S 2 ;. .. ; S N. The servers have di¤erent rates. The service times at each server are independent and exponentially distributed. The objective is to determine the policy which minimizes the average number of customers in the system. We show that any optimal, nonpreemptive policy is of threshold type, i.e., it assigns a customer to server S i , if this server is the fastest server available and the number of customers in the queue is m i or more. The threshold m i may depend on the condition of other (slower) servers at the decision instant. In order to establish the results, we reformulate the optimal control problem as a linear program and use a novel argument based on the structure of the constraint matrix.

On the structure of value functions for threshold policies in queueing models

Journal of Applied Probability, 2003

We study the multi-server queue with Poisson arrivals and identical independent servers with exponentially distributed service times. Customers arriving to the system are admitted or rejected according to a fixed threshold policy. Moreover, the system is subject to holding, waiting, and rejection costs. We give a closed-form expression for the average costs and the value function for this multi-server queue. The result will then be used in a single step of policy iteration in the model where a controller has to route to several finite buffer queues with multiple servers. We numerically show that the improved policy has a close to optimal value.

Structural properties of the optimal resource allocation policy for single-queue systems

Annals of Operations Research, 2013

This paper studies structural properties of the optimal resource allocation policy for singlequeue systems. Jobs arrive at a service facility and are sent one by one to a pool of computing resources for parallel processing. The facility poses a constraint on the maximum expected sojourn time of a job. A central decision maker allocates the servers dynamically to the facility. We consider two models: a limited resource allocation model, where the allocation of resources can only be changed at the start of a new service, and a fully flexible allocation model, where the allocation of resources can also change during a service period. In these two models, the objective is to minimize the average utilization costs whilst satisfying the time constraint. To this end, we cast these optimization problems as Markov decision problems and derive structural properties of the relative value function. We show via dynamic programming that (1) the optimal allocation policy has a work-conservation property, and (2) the optimal number of servers follows a step function with as extreme policy the bang-bang control policy.

Sensitivity Analysis of Markovian Queue with Discouragement, Additional Servers and Threshold Policy

International Journal of System and Software Engineering, 2017

In the present paper, an attempt has been made to study the optimal threshold policy for Markovian queueing model having additional servers along with permanent server. The incorporation of customer's balking and reneging behavior has been done. The customers arrive in Poisson fashion and their service times are exponentially distributed. The first server starts service when N (≥1) or more customers are accumulated and turns off when the system is empty. The (j+1)th (j=1, 2,...., r-1) server turns on when there are Nj+1 customers in the system and will be removed as soon as the number of customers drops to threshold level Nj. We use Laplace transform technique to derive transient probabilities and some other system characteristics such as the expected number of jobs in the system, throughput, and probability that jth (j=1,2,3,…,r) server being busy in rendering the service, etc.. The effects of system parameters on the performance characteristics have been examined by taking numerical illustrations.

Steady-State Analysis of a Flexible Markovian Queue with Server Breakdowns

Entropy, 2019

A flexible single-server queueing system is considered in this paper. The server adapts to the system size by using a strategy where the service provided can be either single or bulk depending on some threshold level c. If the number of customers in the system is less than c, then the server provides service to one customer at a time. If the number of customers in the system is greater than or equal to c, then the server provides service to a group of c customers. The service times are exponential and the service rates of single and bulk service are different. While providing service to either a single or a group of customers, the server may break down and goes through a repair phase. The breakdowns follow a Poisson distribution and the breakdown rates during single and bulk service are different. Also, repair times are exponential and repair rates during single and bulk service are different. The probability generating function and linear operator approaches are used to derive the ...

Optimality of D-Policies for an M/G/1 Queue with a Removable Server

2002

We consider an M/G/1 queue with a removable server. When a customer arrives, the workload becomes known. The cost structure consists of switching costs, running costs, and holding costs per unit time which is a nonnegative nondecreasing right-continuous function of a current workload in the system. We prove an old conjecture that D-policies are optimal for the average cost per unit time criterion. It means that for this criterion there is an optimal policy that either runs the server all the time or switches the server off when the system becomes empty and switches it on when the workload reaches or exceeds some threshold D.