scholarly journals Transient performance analysis of a single server queuing model with retention of reneging customers

2018 ◽  
Vol 28 (3) ◽  
pp. 315-331 ◽  
Author(s):  
Rakesh Kumar ◽  
Sapana Sharma
Keyword(s):  
Performance Analysis ◽  
Probability Generating Function ◽  
Time Dependent ◽  
Function Technique ◽  
Single Server ◽  
Transient Solution ◽  
Transient Performance ◽  
Queuing Model ◽  
Special Cases ◽  
Mean And Variance

In this paper, we study a single server queuing model with retention of reneging customers. The transient solution of the model is derived using probability generating function technique. The time-dependent mean and variance of the model are also obtained. Some important special cases of the model are derived and discussed. Finally, based on the numerical example, the transient performance analysis of the model is performed.

Transient and steady-state analysis of a queuing system having customers’ impatience with threshold

2019 ◽  
Vol 53 (5) ◽  
pp. 1861-1876 ◽  
Author(s):  
Sapana Sharma ◽  
Rakesh Kumar ◽  
Sherif Ibrahim Ammar
Keyword(s):  
Steady State ◽  
Probability Generating Function ◽  
Threshold Value ◽  
Steady State Solution ◽  
Function Technique ◽  
Single Server ◽  
Queuing Model ◽  
Queuing Models ◽  
Special Cases ◽  
Number Of Customers

In many practical queuing situations reneging and balking can only occur if the number of customers in the system is greater than a certain threshold value. Therefore, in this paper we study a single server Markovian queuing model having customers’ impatience (balking and reneging) with threshold, and retention of reneging customers. The transient analysis of the model is performed by using probability generating function technique. The expressions for the mean and variance of the number of customers in the system are obtained and a numerical example is also provided. Further the steady-state solution of the model is obtained. Finally, some important queuing models are derived as the special cases of this model.

An N-policy discrete-time Geo/G/1 queue with modified multiple server vacations and Bernoulli feedback*

2019 ◽  
Vol 53 (2) ◽  
pp. 367-387
Author(s):  
Shaojun Lan ◽  
Yinghui Tang
Keyword(s):  
Discrete Time ◽  
Queue Length ◽  
Length Distribution ◽  
Probability Generating Function ◽  
Renewal Theory ◽  
Function Technique ◽  
Single Server ◽  
Bernoulli Feedback ◽  
Queue Length Distribution ◽  
Special Cases

This paper deals with a single-server discrete-time Geo/G/1 queueing model with Bernoulli feedback and N-policy where the server leaves for modified multiple vacations once the system becomes empty. Applying the law of probability decomposition, the renewal theory and the probability generating function technique, we explicitly derive the transient queue length distribution as well as the recursive expressions of the steady-state queue length distribution. Especially, some corresponding results under special cases are directly obtained. Furthermore, some numerical results are provided for illustrative purposes. Finally, a cost optimization problem is numerically analyzed under a given cost structure.

Interconnected birth and death processes

1968 ◽  
Vol 5 (02) ◽  
pp. 334-349 ◽  
Author(s):  
Prem S. Puri
Keyword(s):  
Limit Theorems ◽  
Probability Generating Function ◽  
Random Variables ◽  
Time Dependent ◽  
Explicit Solutions ◽  
Transition Rates ◽  
Birth And Death Processes ◽  
Death Rates ◽  
Standard Methods ◽  
Special Cases

SummaryTwo cases of multiple linearly interconnected linear birth and death processes are considered. It is found that in general the solution of the Kolmogorov differential equations for the probability generating function (p.g.f)gof the random variables involved is not obtainable by standard methods, although one can obtain moments of the random variables from these equations. A method is considered for obtaining an approximate solution forg.This is based on the introduction of a sequence of stochastic processes such that the sequence {f(n)} of their p.g.f.'s tends togasn → ∞in an appropriate manner. The method is applied to the simple case of two birth and death processes with birth and death rates λiandμi, i =1,2, interconnected linearly with transition rates v andδ(see Figure 2). For this case some limit theorems are established and the probability of ultimate extinction of both the processes is considered. In addition, for the special cases (i) λ1=δ= 0, with the remaining rates time dependent and (ii) λ2=δ= 0, with the remaining rates constant, explicit solutions forgare obtained and studied.

scholarly journals A transient solution to an M/M/1 queue: a simple approach

1987 ◽  
Vol 19 (04) ◽  
pp. 997-998 ◽  
Author(s):  
P. R. Parthasarathy
Keyword(s):  
Queueing System ◽  
Simple Approach ◽  
Time Dependent ◽  
Single Server ◽  
Transient Solution ◽  
Poisson Arrivals ◽  
Service Times ◽  
Time Dependent Solution

A time-dependent solution for the number in a single-server queueing system with Poisson arrivals and exponential service times is derived in a direct way.

scholarly journals Transient Solution of a Single Server Queuing Model with Correlated Reneging Using Runge-Kutta Method

2020 ◽  
Vol 5 (5) ◽  
pp. 886-896
Author(s):  
Rakesh Kumar ◽  
Bhavneet Singh Soodan
Keyword(s):  
Queuing Theory ◽  
Real Life ◽  
System Size ◽  
Single Server ◽  
Kutta Method ◽  
Transient Solution ◽  
Queuing Model ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
Expected Waiting Time

In this paper, the concept of correlated reneging is introduced in queuing theory. The reneging considered so far is dependent on system size, but there are many real life situations where customers may renege due to exogenous factors other than the state of the system. Further, the reneging of customer may induce the other customers to renege at two successive time points. Such reneging is called correlated reneging. An M/M/1/K queuing model with correlated reneging is studied. Runge-Kutta method of fourth order is presented to obtain the transient solution of the model. Some performance measures like expected system size and expected waiting time in the system are studied.

Interconnected birth and death processes

1968 ◽  
Vol 5 (2) ◽  
pp. 334-349 ◽  
Author(s):  
Prem S. Puri
Keyword(s):  
Limit Theorems ◽  
Probability Generating Function ◽  
Random Variables ◽  
Time Dependent ◽  
Explicit Solutions ◽  
Transition Rates ◽  
Birth And Death Processes ◽  
Death Rates ◽  
Standard Methods ◽  
Special Cases

SummaryTwo cases of multiple linearly interconnected linear birth and death processes are considered. It is found that in general the solution of the Kolmogorov differential equations for the probability generating function (p.g.f) g of the random variables involved is not obtainable by standard methods, although one can obtain moments of the random variables from these equations. A method is considered for obtaining an approximate solution for g. This is based on the introduction of a sequence of stochastic processes such that the sequence {f(n)} of their p.g.f.'s tends to g as n → ∞ in an appropriate manner. The method is applied to the simple case of two birth and death processes with birth and death rates λi and μi, i = 1,2, interconnected linearly with transition rates v and δ (see Figure 2). For this case some limit theorems are established and the probability of ultimate extinction of both the processes is considered. In addition, for the special cases (i) λ1 = δ = 0, with the remaining rates time dependent and (ii) λ2 = δ = 0, with the remaining rates constant, explicit solutions for g are obtained and studied.

scholarly journals A single-server Markovian queuing system with discouraged arrivals and retention of reneged customers

2014 ◽  
Vol 24 (1) ◽  
pp. 119-126 ◽  
Author(s):  
Rakesh Kumar ◽  
Kumar Sharma
Keyword(s):  
Negative Impact ◽  
Steady State Solution ◽  
Queuing System ◽  
Point Of View ◽  
Single Server ◽  
Queuing Model ◽  
Queuing Models ◽  
Special Cases ◽  
Retention Of Reneged Customers ◽  
Measures Of Effectiveness

Customer impatience has a very negative impact on the queuing system under investigation. If we talk from business point of view, the firms lose their potential customers due to customer impatience, which affects their business as a whole. If the firms employ certain customer retention strategies, then there are chances that a certain fraction of impatient customers can be retained in the queuing system. A reneged customer may be convinced to stay in the queuing system for his further service with some probability, say q and he may abandon the queue without receiving the service with a probability p(=1? q). A finite waiting space Markovian single-server queuing model with discouraged arrivals, reneging and retention of reneged customers is studied. The steady state solution of the model is derived iteratively. The measures of effectiveness of the queuing model are also obtained. Some important queuing models are derived as special cases of this model.

scholarly journals A feedback queue with additional optional batch service

2014 ◽  
Vol 6 (2) ◽  
Author(s):  
R. Kalayanaraman ◽  
S. Sumathy
Keyword(s):  
Queue Length ◽  
Probability Generating Function ◽  
Arrival Process ◽  
Single Server ◽  
Queuing Model ◽  
Steady State Probability ◽  
Poisson Arrival ◽  
Optional Service ◽  
Essential Service ◽  
Feedback Queue

A single server infinite capacity queuing system with Poisson arrival process along with Bernoulli feedback decision process is considered wherein the server provides two types of service. The first essential service is rendered one by one to all the customers and second optional service is given in batches of fixed size b. For this model the steady state probability generating function for the queue length process has been obtained and average queue length has been found explicitly. Results for particular cases are obtained and some numerical results are presented to test the feasibility of the queuing model.

ON A QUEUING MODEL WITH SERVICE INTERRUPTIONS

2008 ◽  
Vol 22 (4) ◽  
pp. 537-555 ◽  
Author(s):  
Onno Boxma ◽  
Michel Mandjes ◽  
Offer Kella
Keyword(s):  
Queue Length ◽  
Production Systems ◽  
Single Server ◽  
Queuing Model ◽  
Computer Communication ◽  
Polling Models ◽  
Special Cases ◽  
Wide Range ◽  
Workload Distribution ◽  
Almost All

Single-server queues in which the server takes vacations arise naturally as models for a wide range of computer, communication, and production systems. In almost all studies on vacation models, the vacation lengths are assumed to be independent of the arrival, service, workload, and queue length processes. In the present study, we allow the length of a vacation to depend on the length of the previous active period (viz. the period since the previous vacation). Under rather general assumptions regarding the offered work during active periods and vacations, we determine the steady-state workload distribution, both for single and multiple vacations. We conclude by discussing several special cases, including polling models, and relate our findings to results obtained earlier.

scholarly journals A transient solution to an M/M/1 queue: a simple approach

1987 ◽  
Vol 19 (4) ◽  
pp. 997-998 ◽  
Author(s):  
P. R. Parthasarathy
Keyword(s):  
Queueing System ◽  
Simple Approach ◽  
Time Dependent ◽  
Single Server ◽  
Transient Solution ◽  
Poisson Arrivals ◽  
Service Times ◽  
Time Dependent Solution

A time-dependent solution for the number in a single-server queueing system with Poisson arrivals and exponential service times is derived in a direct way.

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