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.

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 Markovian Queueing System with Discouraged Arrivals and Self-Regulatory Servers

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
K. V. Abdul Rasheed ◽  
M. Manoharan
Keyword(s):  
Queueing System ◽  
Queueing Systems ◽  
Single Server ◽  
Expected Number ◽  
Multiple Servers ◽  
Special Cases ◽  
Service Rates ◽  
Expected Waiting Time ◽  
Number Of Customers ◽  
Steady State Probabilities

We consider discouraged arrival of Markovian queueing systems whose service speed is regulated according to the number of customers in the system. We will reduce the congestion in two ways. First we attempt to reduce the congestion by discouraging the arrivals of customers from joining the queue. Secondly we reduce the congestion by introducing the concept of service switches. First we consider a model in which multiple servers have three service ratesμ1,μ2, andμ(μ1≤μ2<μ), say, slow, medium, and fast rates, respectively. If the number of customers in the system exceeds a particular pointK1orK2, the server switches to the medium or fast rate, respectively. For this adaptive queueing system the steady state probabilities are derived and some performance measures such as expected number in the system/queue and expected waiting time in the system/queue are obtained. Multiple server discouraged arrival model having one service switch and single server discouraged arrival model having one and two service switches are obtained as special cases. A Matlab program of the model is presented and numerical illustrations are given.

scholarly journals Improving Queuing System with Limited Resources using TRIZ and Arena Simulation

2020 ◽  
Vol 9 (7) ◽  
pp. 911-919
Keyword(s):  
Waiting Time ◽  
Queueing System ◽  
Queueing Systems ◽  
Research Work ◽  
Arrival Rate ◽  
Time Dependent ◽  
Limited Resources ◽  
Time Interval ◽  
Stationary Time

A university cafeteria is a queueing system characterised by non-stationary time of arrival with limited resources where the arrival rate is time dependent and has different pattern of arrival for different time interval. This means at certain time of the day; the arrival rate is much higher than other time. For a university cafeteria, the arrival rate of customer during the lunchtime is higher and the food (resources) is limited. Non-stationary time dependent queueing systems are not easily modelled mathematically hence such queueing systems are modelled using simulation tools such as ARENA. In order to model a non-stationary time dependent queueing system with limited resources and solve queueing problems using ARENA, researchers have to rely on their knowledge and experience to identify the appropriate parameters of the system and make modifications to these parameters of the system to solve queueing problems by means of trial and error. Hence, this research work explores the potentials of applying a systematic problem solving tool, TRIZ to help users to make better decisions in deriving solutions to improve a non-stationary time dependent queueing system with limited resources. A case study was carried out to minimize the waiting time of the customers at the cafeteria of the Faculty of Engineering, Universiti Putra Malaysia (UPM), which has queueing problems for years during lunchtime. TRIZ was applied in this case study and the results showed that TRIZ can assist researchers to derive a solution model that leads to shorter waiting time without incurring additional cost and resources.

Conditioned Limit Theorems for the Difference of Waiting Time and Queue Length

1994 ◽  
Vol 26 (01) ◽  
pp. 242-257
Author(s):  
Władysław Szczotka ◽  
Krzysztof Topolski
Keyword(s):  
Waiting Time ◽  
Queue Length ◽  
Limit Theorems ◽  
Queueing System ◽  
Queueing Systems ◽  
Traffic Intensity ◽  
Virtual Waiting Time ◽  
The Difference ◽  
Scheme Of Series ◽  
Brownian Meander

Consider the GI/G/1 queueing system with traffic intensity 1 and let wk and lk denote the actual waiting time of the kth unit and the number of units present in the system at the kth arrival including the kth unit, respectively. Furthermore let τ denote the number of units served during the first busy period and μ the intensity of the service. It is shown that as k →∞, where a is some known constant, , , and are independent, is a Brownian meander and is a Wiener process. A similar result is also given for the difference of virtual waiting time and queue length processes. These results are also extended to a wider class of queueing systems than GI/G/1 queues and a scheme of series of queues.

Conditioned Limit Theorems for the Difference of Waiting Time and Queue Length

1994 ◽  
Vol 26 (1) ◽  
pp. 242-257
Author(s):  
Władysław Szczotka ◽  
Krzysztof Topolski
Keyword(s):  
Waiting Time ◽  
Queue Length ◽  
Limit Theorems ◽  
Queueing System ◽  
Queueing Systems ◽  
Traffic Intensity ◽  
Virtual Waiting Time ◽  
The Difference ◽  
Scheme Of Series ◽  
Brownian Meander

Consider the GI/G/1 queueing system with traffic intensity 1 and let wk and lk denote the actual waiting time of the kth unit and the number of units present in the system at the kth arrival including the kth unit, respectively. Furthermore let τ denote the number of units served during the first busy period and μ the intensity of the service. It is shown that as k →∞, where a is some known constant, , , and are independent, is a Brownian meander and is a Wiener process. A similar result is also given for the difference of virtual waiting time and queue length processes. These results are also extended to a wider class of queueing systems than GI/G/1 queues and a scheme of series of queues.

scholarly journals Neutrosophic Event-Based Queueing Model

2020 ◽  
pp. 48-55
Author(s):  
Mohamed Bisher Zeina ◽  
Keyword(s):  
Performance Measures ◽  
Waiting Time ◽  
Communication Systems ◽  
Queueing Systems ◽  
Real Life ◽  
Main Tool ◽  
Expected Number ◽  
Mean Waiting Time ◽  
Event Based ◽  
Number Of Customers

In this paper we have defined the concept of neutrosophic queueing systems and defined its neutrosophic performance measures. An important application of neutrosophic logic in queueing systems we face in real life were discussed, that is the neutrosophic events accuring times, because of its wide applications in networking and simulating communication systems specialy when probability distribution is not known, and because it’s more realistic to consider and to not ignore the imprecise events times. Event-based table of a neutrosophic queueing system was presented and its neutrosophic performance measures were derived, i.e. neutrosophic mean waiting time in queue, neutrosophic mean waiting time in system, neutrosophic expected number of customers in queue and neutrosophic expected number of customers in system. Neutrosophic Little’s Formulas (NLF) were also defined which is a main tool in queueing systems problems to make it easier finding performance measures from each other.

Some analyses on the control of queues using level crossings of regenerative processes

1980 ◽  
Vol 17 (3) ◽  
pp. 814-821 ◽  
Author(s):  
J. G. Shanthikumar
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Queueing Systems ◽  
Density Functions ◽  
Waiting Time Distribution ◽  
Regenerative Processes ◽  
Stieltjes Transform ◽  
Expected Number ◽  
Control Of Queues ◽  
Special Case

Some properties of the number of up- and downcrossings over level u, in a special case of regenerative processes are discussed. Two basic relations between the density functions and the expected number of upcrossings of this process are derived. Using these reults, two examples of controlled M/G/1 queueing systems are solved. Simple relations are derived for the waiting time distribution conditioned on the phase of control encountered by an arriving customer. The Laplace-Stieltjes transform of the distribution function of the waiting time of an arbitrary customer is also derived for each of these two examples.

scholarly journals Applications of Queuing Theory in Quantitative Business Analysis

2020 ◽  
pp. 253-257
Keyword(s):  
Quantitative Analysis ◽  
Waiting Time ◽  
Queuing Theory ◽  
Single Channel ◽  
Analysis Approach ◽  
Single Server ◽  
Expected Number ◽  
Queuing Model ◽  
Infinite Population ◽  
Expected Waiting Time

Queuing Theory provides the system of applications in many sectors in life cycle. Queuing Structure and basic components determination is computed in queuing model simulation process. Distributions in Queuing Model can be extracted in quantitative analysis approach. Differences in Queuing Model Queue discipline, Single and Multiple service station with finite and infinite population is described in Quantitative analysis process. Basic expansions of probability density function, Expected waiting time in queue, Expected length of Queue, Expected size of system, probability of server being busy, and probability of system being empty conditions can be evaluated in this quantitative analysis approach. Probability of waiting ‘t’ minutes or more in queue and Expected number of customer served per busy period, Expected waiting time in System are also computed during the Analysis method. Single channel model with infinite population is used as most common case of queuing problems which involves the single channel or single server waiting line. Single Server model with finite population in test statistics provides the Relationships used in various applications like Expected time a customer spends in the system, Expected waiting time of a customer in the queue, Probability that there are n customers in the system objective case, Expected number of customers in the system

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.

On a property of a refusals stream

1997 ◽  
Vol 34 (03) ◽  
pp. 800-805 ◽  
Author(s):  
Vyacheslav M. Abramov
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Busy Period ◽  
Elementary Proof ◽  
Queueing System ◽  
Queueing Systems ◽  
Single Server ◽  
Wide Family

This paper consists of two parts. The first part provides a more elementary proof of the asymptotic theorem of the refusals stream for an M/GI/1/n queueing system discussed in Abramov (1991a). The central property of the refusals stream discussed in the second part of this paper is that, if the expectations of interarrival and service time of an M/GI/1/n queueing system are equal to each other, then the expectation of the number of refusals during a busy period is equal to 1. This property is extended for a wide family of single-server queueing systems with refusals including, for example, queueing systems with bounded waiting time.

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