The effect of prior assumptions on the expected cost of m/m/1 queueing systems with unknown arrival rate (original) (raw)

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.

A queueing model with the bulk arrival rate depending upon the nature of service

Applied Mathematical Modelling, 1996

In this paper we study the time-dependent analysis of a limited capacity queueing model with the bulk arrival rate depending upon the nature of service available in the system. The customers arrive in the system in batches of size x, which is a random variable, and the service consists of two stages, one is essential (first stage) while the other may be inessential. The decision to offer the inessential service depends upon the size of the system. However, if this inessential service is temporarily suspended, the arrival rate of the customers decreases. Laplace transforms (in time) of the different probability generating functions describing the system size under various conditions of service and the expected system size are derived. Steady-state results consequently follow.

An M/M/c queue with queueing-time dependent service rates

2021

Recent studies indicate that in many situations service times are affected by the experienced queueing delay of the particular customer. This effect has been detected in different areas, such as health care, call centers and telecommunication networks. In this paper we present a methodology to analyze a model having this property. The specific model is an M/M/c queue in which any customer may be tagged at her arrival time if her queueing time will be above a certain fixed threshold. All tagged customers are then served at a given rate that may differ from the rate used for the non-tagged customers. We show how it is possible to model the virtual queueing time of this queueing system by a specific Markov chain. Then, solving the corresponding balance equations, we give a recursive solution to compute the stationary distribution, which involves a mixture of exponential terms. Using numerical experiments, we demonstrate that the differences in service rates can have a crucial impact on...

Measuring the Cost for Some Single Channel Waiting Line Models

Asian Journal of Probability and Statistics, 2018

Queuing models applications are centered on the question of finding the ideal level of services, waiting times and queue lengths. The aim of this study is to measure the cost for three models and compare the cost for the three single channel waiting line models instead of finding the ideal level of services, waiting times and queue lengths which calculated in many studies. Each model depends on two important parameters arrival rate (λ) and service rate (μ) which followed different distributions. The cost for the three single channel waiting line models is calculated when arrival rate (λ) is followed Poisson distribution and service rate (μ) is followed different distributions. The objective for the waiting line models is to minimize total expected costs by minimize the sum of service costs and waiting costs. Therefore, the study concerned with changing the distribution of the service rate (μ) and examining its impact on cost. This choice was made to emphasize the basic idea of the study (there is a relationship between the service rate distribution and the cost). The study results showed that there is a relationship between the service rate distribution and the cost.

Cost and Profit Analysis of M / Ek / 1 Queueing System with Removable Service Station

2008

In this paper, an attempt is made to investigate cost and profit analysis of an M/Ek /1 queuing model with a removable service station under N-policy and steady state conditions. We construct the total expected cost function per unit time for two different cases. In first case, we analyze the total expected cost of the system under the condition when the system is turned off and turned on and in second case, we discuss the total expected cost of the system with utilization of idle time and finally total optimal costs in both of cases have been obtained. The notion of the total revenue is introduced to analyze the total profit of the system with respect to the total cost of the system. Finally, observations on TOC and TEP have been drawn with the help of numerical tables and graphic representations.

A queueing system with decomposed service and inventoried preliminary services

Applied Mathematical Modelling

We study a single-server queue in which the service consists of two independent stages. The first stage is generic and can be performed even in the absence of customers, whereas the second requires the customer to be present. When the system is empty of customers, the server produces an inventory of first-stage ('preliminary') services (denoted PSs), which is used to reduce customers' overall sojourn times. We formulate and analyze the queueing-inventory system and derive its steady-state probabilities by using the matrix geometric method, which is based on calculating the so called rate matrix R. It is shown that the system's stability is not affected by the production rate of PSs, and that there are cases in which utilizing the server's idle time to produce PSs actually increases the fraction of time during which the server is dormant. A significant contribution is the derivation of an explicit expression of R , whose entries are written in terms of Catalan numbers. This type of result is rare in the literature and enables large-scale problems to be solved with low computational effort. Furthermore, by utilizing Laplace-Stieltjes transform and its inverse, we obtain the distribution function of customers' sojourn time. Finally, based on the probabilistic study, we carry out an economic analysis using a practical example from the fast food industry.

On the optimal control of loss probability and profit in a GI/C-BMSP/1/N queueing system

OPSEARCH

This paper deals with a single server queueing system where the waiting space is limited. The server serves the customer in batches. The arrival process is considered to be renewal type and the services are considered to be correlated which has been presented by a continuous-time batch Markovian service process (C-BMSP). Distribution of the system length at pre-arrival instant of a customer and at an arbitrary-epoch have been determined for this queueing system. These probability distributions have been used for obtaining the blocking probability of an arbitrary customer, expected system-length, expected waiting time of an arbitrary customer in the system, and several other important performance measures. This model may find application in queueing systems involving inventory where delay in demand may lead to perishing of goods due to long wait in the system. Also, a profit function has been derived for such a queueing model to maximize the profit from the system for certain model parameters. Finally, assuming that the inter-arrival time follows phase-type distribution, a few numerical examples have been presented in the form of graphs and tables.

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.