THE ALLOCATION OF CUSTOMERS IN A DISCRETE-TIME MULTI-SERVER QUEUEING SYSTEM

2010 ◽  
Vol 27 (06) ◽  
pp. 649-667 ◽  
Author(s):  
WEI SUN ◽  
NAISHUO TIAN ◽  
SHIYONG LI
Keyword(s):  
Performance Measures ◽  
Discrete Time ◽  
Queueing System ◽  
Arrival Rate ◽  
Allocation Problem ◽  
Optimal Outcome ◽  
The Subject ◽  
Multi Server ◽  
Theoretical Results

This paper, analyzes the allocation problem of customers in a discrete-time multi-server queueing system and considers two criteria for routing customers' selections: equilibrium and social optimization. As far as we know, there is no literature concerning the discrete-time multi-server models on the subject of equilibrium behaviors of customers and servers. Comparing the results of customers' distribution at the servers under the two criteria, we show that the servers used in equilibrium are no more than those used in the socially optimal outcome, that is, the individual's decision deviates from the socially preferred one. Furthermore, we also clearly show the mutative trend of several important performance measures for various values of arrival rate numerically to verify the theoretical results.

A note on the convexity of performance measures of M/M/c queueing systems

1983 ◽  
Vol 20 (04) ◽  
pp. 920-923 ◽  
Author(s):  
Hau Leung Lee ◽  
Morris A. Cohen
Keyword(s):  
Performance Measures ◽  
Waiting Time ◽  
Queueing System ◽  
Queueing Systems ◽  
Arrival Rate ◽  
Control Problems ◽  
Traffic Intensity ◽  
Expected Number ◽  
Expected Waiting Time

Convexity of performance measures of queueing systems is important in solving control problems of multi-facility systems. This note proves that performance measures such as the expected waiting time, expected number in queue, and the Erlang delay formula are convex with respect to the arrival rate or the traffic intensity of the M/M/c queueing system.

scholarly journals A discrete-time queueing system with changes in the vacation times

2016 ◽  
Vol 26 (2) ◽  
pp. 379-390 ◽  
Author(s):  
Ivan Atencia
Keyword(s):  
Performance Measures ◽  
Discrete Time ◽  
Busy Period ◽  
Queueing System ◽  
Probability Generating Function ◽  
Service Discipline ◽  
Numerical Examples ◽  
Extensive Analysis ◽  
The One ◽  
Steady State Performance

Abstract This paper considers a discrete-time queueing system in which an arriving customer can decide to follow a last come first served (LCFS) service discipline or to become a negative customer that eliminates the one at service, if any. After service completion, the server can opt for a vacation time or it can remain on duty. Changes in the vacation times as well as their associated distribution are thoroughly studied. An extensive analysis of the system is carried out and, using a probability generating function approach, steady-state performance measures such as the first moments of the busy period of the queue content and of customers delay are obtained. Finally, some numerical examples to show the influence of the parameters on several performance characteristics are given.

A DISCRETE-TIME Geo/G/1 RETRIAL QUEUE WITH SERVER BREAKDOWNS

2006 ◽  
Vol 23 (02) ◽  
pp. 247-271 ◽  
Author(s):  
IVAN ATENCIA ◽  
PILAR MORENO
Keyword(s):  
Steady State ◽  
Performance Measures ◽  
Discrete Time ◽  
Continuous Time ◽  
Queueing System ◽  
Retrial Queue ◽  
System Size ◽  
Stochastic Decomposition ◽  
Recursive Procedure ◽  
Server Breakdown

This paper discusses a discrete-time Geo/G/1 retrial queue with the server subject to breakdowns and repairs. The customer just being served before server breakdown completes his remaining service when the server is fixed. The server lifetimes are assumed to be geometrical and the server repair times are arbitrarily distributed. We study the Markov chain underlying the considered queueing system and present its stability condition as well as some performance measures of the system in steady-state. Then, we derive a stochastic decomposition law and as an application we give bounds for the proximity between the steady-state distributions of our system and the corresponding system without retrials. Also, we introduce the concept of generalized service time and develop a recursive procedure to obtain the steady-state distributions of the orbit and system size. Finally, we prove the convergence to the continuous-time counterpart and show some numerical results.

scholarly journals Priority Multi-Server Queueing System with Heterogeneous Customers

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1501
Author(s):  
Valentina Klimenok ◽  
Alexander Dudin ◽  
Vladimir Vishnevsky
Keyword(s):  
Information Transmission ◽  
Performance Measures ◽  
Queueing System ◽  
Real Life ◽  
Telecommunication Networks ◽  
Arrival Process ◽  
Phase Type Distribution ◽  
Arrival Times ◽  
Process Measurements ◽  
Multi Server

In this paper, we analyze a multi-server queueing system with heterogeneous customers that arrive according to a marked Markovian arrival process. Customers of two types differ in priorities and parameters of phase type distribution of their service time. The queue under consideration can be used to model the processes of information transmission in telecommunication networks in which often the flow of information is the superposition of several types of flows with correlation of inter-arrival times within each flow and cross-correlation. We define the process of information transmission as the multi-dimensional Markov chain, derive the generator of this chain and compute its stationary distribution. Expressions for computation of various performance measures of the system, including the probabilities of loss of customers of different types, are presented. Output flow from the system is characterized. The presented numerical results confirm the high importance of account of correlation in the arrival process. The values of important performance measures for the systems with the correlated arrival process are essentially different from the corresponding values for the systems with the stationary Poisson arrival process. Measurements in many real world systems show poor approximation of real flows by such an arrival process. However, this process is still popular among the telecommunication engineers due to the evident existing gap between the needs of adequately modeling the real life systems and the current state of the theory of algorithmic methods of queueing theory.

scholarly journals A Discrete-Time UnreliableGeo/G/1Retrial Queue with Balking Customers, Second Optional Service, and General Retrial Times

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Feng Zhang ◽  
Zhifeng Zhu
Keyword(s):  
Performance Measures ◽  
Discrete Time ◽  
Queueing System ◽  
System Size ◽  
Stochastic Decomposition ◽  
State Behavior ◽  
Retrial Queueing System ◽  
Retrial Queueing ◽  
Optional Service ◽  
Second Optional Service

This paper deals with the steady-state behavior of a discrete-time unreliableGeo/G/1retrial queueing system with balking customers and second optional service. The server may break down randomly while serving the customers. If the server breaks down, the server is sent to be repaired immediately. We analyze the Markov chain underlying the considered system and its ergodicity condition. Then, we obtain some performance measures based on the generating functions. Moreover, a stochastic decomposition result of the system size is investigated. Finally, some numerical examples are provided to illustrate the effect of some parameters on main performance measures of the system.

scholarly journals Performance Analysis of a Discrete-Time Queue with Working Breakdowns and Searching for the Optimum Service Rate in Working Breakdown Period

2017 ◽  
Vol 5 (2) ◽  
pp. 176-192
Author(s):  
Shaojun Lan ◽  
Yinghui Tang
Keyword(s):  
Performance Measures ◽  
Generating Function ◽  
Discrete Time ◽  
Queueing System ◽  
Operating Cost ◽  
Probability Generating Function ◽  
Repair Process ◽  
Function Technique ◽  
Service Rate ◽  
The Stability

Abstract This paper deals with a discrete-time Geo/Geo/1 queueing system with working breakdowns in which customers arrive at the system in variable input rates according to the states of the server. The server may be subject to breakdowns at random when it is in operation. As soon as the server fails, a repair process immediately begins. During the repair period, the defective server still provides service for the waiting customers at a lower service rate rather than completely stopping service. We analyze the stability condition for the considered system. Using the probability generating function technique, we obtain the probability generating function of the steady-state queue size distribution. Also, various important performance measures are derived explicitly. Furthermore, some numerical results are provided to carry out the sensitivity analysis so as to illustrate the effect of different parameters on the system performance measures. Finally, an operating cost function is formulated to model a computer system and the parabolic method is employed to numerically find the optimum service rate in working breakdown period.

A note on the convexity of performance measures of M/M/c queueing systems

1983 ◽  
Vol 20 (4) ◽  
pp. 920-923 ◽  
Author(s):  
Hau Leung Lee ◽  
Morris A. Cohen
Keyword(s):  
Performance Measures ◽  
Waiting Time ◽  
Queueing System ◽  
Queueing Systems ◽  
Arrival Rate ◽  
Control Problems ◽  
Traffic Intensity ◽  
Expected Number ◽  
Expected Waiting Time

Convexity of performance measures of queueing systems is important in solving control problems of multi-facility systems. This note proves that performance measures such as the expected waiting time, expected number in queue, and the Erlang delay formula are convex with respect to the arrival rate or the traffic intensity of the M/M/c queueing system.

Optimal design of a multi-server queueing system with delay information

2016 ◽  
Vol 116 (1) ◽  
pp. 147-169 ◽  
Author(s):  
Miao Yu ◽  
Jun Gong ◽  
Jiafu TANG
Keyword(s):  
Optimal Design ◽  
Performance Measures ◽  
Call Center ◽  
Queueing System ◽  
Call Centers ◽  
Queueing Systems ◽  
Staffing Level ◽  
Content Type ◽  
Multi Server ◽  
Delay Information

Purpose – The purpose of this paper is to provide a framework for the optimal design of queueing systems of call centers with delay information. The main decisions in the design of such systems are the number of servers, the appropriate control to announce delay anticipated. Design/methodology/approach – This paper models a multi-server queueing system as an M/M/S+M queue with customer reactions. Based on customer psychology in waiting experiences, a number of different service-level definitions are structured and the explicit computation of their performance measures is performed. This paper characterizes the level of satisfaction with delay information to modulate customer reactions. Optimality is defined as the number of agents that maximize revenues net of staffing costs. Findings – Numerical studies show that the solutions to optimal design of staffing levels and delay information exhibit interesting differences, especially U-shaped curve for optimal staffing level. Experiments show how call center managers can determine economically optimal anticipated delay and number of servers so that they could control the trade-off between revenue loss and customer satisfaction. Originality/value – Many results that pertain to announcing delay information, customer reactions, and links to satisfaction with delay information have not been established in previous studies, however, this paper analytically characterizes these performance measures for staffing call centers.

scholarly journals A discrete-time system with service control and repairs

2014 ◽  
Vol 24 (3) ◽  
pp. 471-484 ◽  
Author(s):  
Ivan Atencia
Keyword(s):  
Performance Measures ◽  
Discrete Time ◽  
Generating Functions ◽  
Continuous Time ◽  
Queueing System ◽  
Discrete Time System ◽  
Time System ◽  
The Third ◽  
Special Cases ◽  
Number Of Customers

Abstract This paper discusses a discrete-time queueing system with starting failures in which an arriving customer follows three different strategies. Two of them correspond to the LCFS (Last Come First Served) discipline, in which displacements or expulsions of customers occur. The third strategy acts as a signal, that is, it becomes a negative customer. Also examined is the possibility of failures at each service commencement epoch. We carry out a thorough study of the model, deriving analytical results for the stationary distribution. We obtain the generating functions of the number of customers in the queue and in the system. The generating functions of the busy period as well as the sojourn times of a customer at the server, in the queue and in the system, are also provided. We present the main performance measures of the model. The versatility of this model allows us to mention several special cases of interest. Finally, we prove the convergence to the continuous-time counterpart and give some numerical results that show the behavior of some performance measures with respect to the most significant parameters of the system

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