scholarly journals Equilibrium Customer Strategies in the Queue with Threshold Policy and Setup Times

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Peishu Chen ◽  
Wenhui Zhou ◽  
Yongwu Zhou
Keyword(s):  
Performance Measures ◽  
Setup Time ◽  
System Size ◽  
Social Benefit ◽  
Setup Times ◽  
Mixed Strategies ◽  
Threshold Policy ◽  
Numerical Examples ◽  
Equilibrium Behavior ◽  
Number Of Customers

We consider the equilibrium behavior of customers in theM/M/1queue withNpolicy and setup times. The server is shut down once the system is empty and begins an exponential setup time to start service again when the number of customers in the system accumulates the thresholdN  (N≥1). We consider the equilibrium threshold strategies for fully observable case and mixed strategies for fully unobservable case, respectively. We get various performance measures of the system and investigate some numerical examples of system size, social benefit, and expected cost function per unit time for the two different cases under equilibrium customer strategies.

A single server Markovian queuing system with limited buffer and reverse balking

2021 ◽  
Vol 12 (7) ◽  
pp. 1774-1784
Author(s):  
Girin Saikia ◽  
Amit Choudhury
Keyword(s):  
Steady State ◽  
Performance Measures ◽  
Mutual Fund ◽  
System Size ◽  
Queuing System ◽  
Single Server ◽  
Insurance Company ◽  
Number Of Customers ◽  
Steady State Probabilities

The phenomena are balking can be said to have been observed when a customer who has arrived into queuing system decides not to join it. Reverse balking is a particular type of balking wherein the probability that a customer will balk goes down as the system size goes up and vice versa. Such behavior can be observed in investment firms (insurance company, Mutual Fund Company, banks etc.). As the number of customers in the firm goes up, it creates trust among potential investors. Fewer customers would like to balk as the number of customers goes up. In this paper, we develop an M/M/1/k queuing system with reverse balking. The steady-state probabilities of the model are obtained and closed forms of expression of a number of performance measures are derived.

The disasters queue with working breakdowns and impatient customers

2020 ◽  
Vol 54 (3) ◽  
pp. 815-825
Author(s):  
Mian Zhang ◽  
Shan Gao
Keyword(s):  
Size Distribution ◽  
Performance Measures ◽  
Sojourn Time ◽  
Queue Length ◽  
Slow Rate ◽  
System Size ◽  
Impatient Customers ◽  
Numerical Examples ◽  
The Mean ◽  
Mean Queue Length

We consider the M/M/1 queue with disasters and impatient customers. Disasters only occur when the main server being busy, it not only removes out all present customers from the system, but also breaks the main server down. When the main server is down, it is sent for repair. The substitute server serves the customers at a slow rate(working breakdown service) until the main server is repaired. The customers become impatient due to the working breakdown. The system size distribution is derived. We also obtain the mean queue length of the model and mean sojourn time of a tagged customer. Finally, some performance measures and numerical examples are presented.

Confidence Intervals for Performance Measures of M/M/1 Queue with Constant Retrial Policy

2015 ◽  
Vol 32 (06) ◽  
pp. 1550046
Author(s):  
Dmitry Efrosinin ◽  
Anastasia Winkler ◽  
Pinzger Martin
Keyword(s):  
Performance Measures ◽  
Queueing System ◽  
Retrial Queue ◽  
Threshold Level ◽  
Stationary Regime ◽  
Parametric Estimation ◽  
Threshold Policy ◽  
System Parameters ◽  
Number Of Customers ◽  
The Given

We consider the problem of estimation and confidence interval construction of a Markovian controllable queueing system with unreliable server and constant retrial policy. For the fully observable system the standard parametric estimation technique is used. The arrived customer finding a free server either gets service immediately or joins a retrial queue. The customer at the head of the retrial queue is allowed to retry for service. When the server is busy, it is subject to breakdowns. In a failed state the server can be repaired with respect to the threshold policy: the repair starts when the number of customers in the system reaches a fixed threshold level. To obtain the estimates for the system parameters, performance measures and optimal threshold level we analyze the system in a stationary regime. The performance measures including average cost function for the given cost structure are presented in a closed matrix form.

ON THE VARIANCES OF SYSTEM SIZE AND SOJOURN TIME IN A DISCRETE-TIME DAR(1)/D/1 QUEUE

2011 ◽  
Vol 25 (4) ◽  
pp. 519-535 ◽  
Author(s):  
Daniel Wei-Chung Miao ◽  
Hung Chen
Keyword(s):  
Closed Form ◽  
Recurrence Relation ◽  
Performance Measures ◽  
Discrete Time ◽  
Sojourn Time ◽  
Blow Up ◽  
System Size ◽  
Batch Size ◽  
Numerical Examples ◽  
Cross Terms

We consider a discrete-time DAR(1)/D/1 queue and provide an analysis on the variances of both its system size and sojourn time. Our approach is simple, but the results are nice, as these variances are found in closed form. We first establish the relation between these variances, based on which we then use the conditioning technique to analyze the expected cross terms that come from its system recurrence relation. The closed-form results allow us to explicitly examine the effect from the batch size distribution and the autocorrelation parameter p. It is observed that as p grows toward 1, the standard deviations of the two performance measures will blow up in same asymptotic order of O(1/(1−p)) as their means. These are demonstrated through numerical examples.

Randomized control of arrivals in a finite-buffer GI/M/1 system with starting failures

2020 ◽  
Vol 54 (2) ◽  
pp. 351-367 ◽  
Author(s):  
Dong-Yuh Yang ◽  
Jau-Chuan Ke ◽  
Chia-Huang Wu
Keyword(s):  
Performance Measures ◽  
Cost Model ◽  
System Size ◽  
Single Server ◽  
Finite Buffer ◽  
Optimal Capacity ◽  
Startup Time ◽  
Startup Process ◽  
Randomized Control ◽  
Number Of Customers

This paper proposes a randomized policy for the control of arrivals in a finite-buffer GI/M/1 system with a single server. When the capacity of system is full, no new arrivals are permitted to enter the system. When the number of customers in the system decreases to threshold F, a new arriving customer is allowed to join the system with probability p. The system requires an exponential startup time before allowing customers to enter the system. The startup process may not be successful and is then restarted once again. Using the supplementary variable technique in a recursive process, we obtain the stationary distribution of the system size. Various performance measures of the system are developed. We also create a cost model based on the system performance measures and cost elements. The optimal threshold, optimal capacity and optimal startup rate of the system are determined to minimize the expected cost per unit time. Finally, we provide numerical examples to conduct a sensitivity analysis.

scholarly journals Analysis of Repairable Geo/G/1 Queues with Negative Customers

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Doo Ho Lee ◽  
Kilhwan Kim
Keyword(s):  
Mathematical Model ◽  
Performance Measures ◽  
Generating Functions ◽  
Time Distribution ◽  
System Size ◽  
Repair Process ◽  
Negative Customers ◽  
Numerical Examples ◽  
Sojourn Time Distribution ◽  
The Mathematical Model

We consider discrete-time Geo/G/1 queues with negative customers and a repairable server. The server is subject to failure due to a negative customer arrival. As soon as a negative customer arrives at a system, the server fails and one positive (ordinary) customer is forced to leave. At a failure instant, the server is turned off and the repair process immediately begins. We construct the mathematical model and present the probability generating functions of the system size distribution and the FCFS sojourn time distribution. Finally, some numerical examples are given to show the influence of negative customer arrival on the performance measures of the system.

scholarly journals On the GI/M/1 Queue with Vacations and Multiple Service Phases

2017 ◽  
Vol 2017 ◽  
pp. 1-14
Author(s):  
Jianjun Li ◽  
Liwei Liu
Keyword(s):  
Performance Measures ◽  
Waiting Time ◽  
Solution Method ◽  
System Size ◽  
Mean Waiting Time ◽  
The Matrix ◽  
Matrix Geometric Solution ◽  
Semi Markov Process ◽  
Stationary Waiting Time ◽  
Number Of Customers

This paper considers a GI/M/1 queue with vacations and multiple service phases. Whenever the system becomes empty, the server takes a vacation, causing the system to move to vacation phase 0. If the server returns from a vacation to find no customer waiting, another vacation begins. Otherwise, the system jumps from phase 0 to some service phase i with probability qi,  i=1,2,…,N. Using the matrix geometric solution method and semi-Markov process, we obtain the distributions of the stationary system size at both arrival and arbitrary epochs. The distribution of the stationary waiting time of an arbitrary customer is also derived. In addition, we present some performance measures such as mean waiting time of an arbitrary customer, mean length of the type-i cycle, and mean number of customers in the system at the end of phase 0. Finally, some numerical examples are presented.

Tollbooth tandem queues with infinite homogeneous servers

2015 ◽  
Vol 52 (4) ◽  
pp. 941-961 ◽  
Author(s):  
Xiuli Chao ◽  
Qi-Ming He ◽  
Sheldon Ross
Keyword(s):  
Performance Measures ◽  
System Performance ◽  
Stochastic Ordering ◽  
Tandem Queues ◽  
Poisson Arrival ◽  
Arrival Processes ◽  
Poisson Arrivals ◽  
Number Of Customers ◽  
Steady State Solutions ◽  
Transient And Steady State

In this paper we analyze a tollbooth tandem queueing problem with an infinite number of servers. A customer starts service immediately upon arrival but cannot leave the system before all customers who arrived before him/her have left, i.e. customers depart the system in the same order as they arrive. Distributions of the total number of customers in the system, the number of departure-delayed customers in the system, and the number of customers in service at time t are obtained in closed form. Distributions of the sojourn times and departure delays of customers are also obtained explicitly. Both transient and steady state solutions are derived first for Poisson arrivals, and then extended to cases with batch Poisson and nonstationary Poisson arrival processes. Finally, we report several stochastic ordering results on how system performance measures are affected by arrival and service processes.

scholarly journals A M/M/1 Queueing Network with Classical Retrial Policy

2021 ◽  
Vol 12 (1S) ◽  
pp. 329-337
Author(s):  
S. Shanmugasundaram, Et. al.
Keyword(s):  
Steady State ◽  
Queue Length ◽  
Queueing Network ◽  
State Probability ◽  
Queueing Model ◽  
Steady State Probability ◽  
Numerical Examples ◽  
System Length ◽  
Number Of Customers ◽  
Little's Formula

In this paper we study the M/M/1 queueing model with retrial on network. We derive the steady state probability of customers in the network, the average number of customers in the all the three nodes in the system, the queue length, system length using little’s formula. The particular case is derived (no retrial). The numerical examples are given to test the correctness of the model.

Analysis of the M x/G/1 queue by N-policy and multiple vacations

1994 ◽  
Vol 31 (02) ◽  
pp. 476-496
Author(s):  
Ho Woo Lee ◽  
Soon Seok Lee ◽  
Jeong Ok Park ◽  
K. C. Chae
Keyword(s):  
Performance Measures ◽  
Queue Length ◽  
Time Distribution ◽  
Queueing System ◽  
Random Variables ◽  
System Size ◽  
The Other ◽  
Waiting Time Distribution ◽  
Multiple Vacations ◽  
Linear Cost

We consider an Mx /G/1 queueing system with N-policy and multiple vacations. As soon as the system empties, the server leaves for a vacation of random length V. When he returns, if the queue length is greater than or equal to a predetermined value N(threshold), the server immediately begins to serve the customers. If he finds less than N customers, he leaves for another vacation and so on until he finally finds at least N customers. We obtain the system size distribution and show that the system size decomposes into three random variables one of which is the system size of ordinary Mx /G/1 queue. The interpretation of the other random variables will be provided. We also derive the queue waiting time distribution and other performance measures. Finally we derive a condition under which the optimal stationary operating policy is achieved under a linear cost structure.

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