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.

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.

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.

The steady-state solution of the M/K2/m queue

1980 ◽  
Vol 12 (03) ◽  
pp. 799-823
Author(s):  
Per Hokstad
Keyword(s):  
Steady State ◽  
Queue Length ◽  
Simple Formula ◽  
Length Distribution ◽  
Steady State Solution ◽  
Queue Length Distribution ◽  
Server Queue ◽  
The Mean ◽  
Mean Queue Length ◽  
The Many

The many-server queue with service time having rational Laplace transform of order 2 is considered. An expression for the asymptotic queue-length distribution is obtained. A relatively simple formula for the mean queue length is also found. A few numerical results on the mean queue length and on the probability of having to wait are given for the case of three servers. Some approximations for these quantities are also considered.

The steady-state solution of the M/K2/m queue

1980 ◽  
Vol 12 (3) ◽  
pp. 799-823 ◽  
Author(s):  
Per Hokstad
Keyword(s):  
Steady State ◽  
Queue Length ◽  
Simple Formula ◽  
Length Distribution ◽  
Steady State Solution ◽  
Queue Length Distribution ◽  
Server Queue ◽  
The Mean ◽  
Mean Queue Length ◽  
The Many

The many-server queue with service time having rational Laplace transform of order 2 is considered. An expression for the asymptotic queue-length distribution is obtained. A relatively simple formula for the mean queue length is also found. A few numerical results on the mean queue length and on the probability of having to wait are given for the case of three servers. Some approximations for these quantities are also considered.

Queues with time-dependent arrival rates. III — A mild rush hour

1968 ◽  
Vol 5 (3) ◽  
pp. 591-606 ◽  
Author(s):  
G. F. Newell
Keyword(s):  
Queueing Theory ◽  
Queue Length ◽  
Arrival Rate ◽  
Time Dependent ◽  
Service Rate ◽  
Diffusion Approximations ◽  
Dimensionless Form ◽  
Queue Lengths ◽  
The Mean ◽  
Mean Queue Length

The arrival rate of customers to a service facility is assumed to have the form λ(t) = λ(0) — βt2 for some constant β. Diffusion approximations show that for λ(0) sufficiently close to the service rate μ, the mean queue length at time 0 is proportional to β–1/5. A dimensionless form of the diffusion equation is evaluated numerically from which queue lengths can be evaluated as a function of time for all λ(0) and β. Particular attention is given to those situations in which neither deterministic queueing theory nor equilibrium stochastic queueing theory apply.

scholarly journals Analysis of a Single-Sever Queue with Disasters and Repairs Under Bernoulli Vacation Schedule

2016 ◽  
Vol 4 (6) ◽  
pp. 547-559
Author(s):  
Jingjing Ye ◽  
Liwei Liu ◽  
Tao Jiang
Keyword(s):  
Performance Measures ◽  
Generating Functions ◽  
Sojourn Time ◽  
Time Distribution ◽  
Busy Period ◽  
Balance Equations ◽  
Sojourn Time Distribution ◽  
The Mean ◽  
System Length ◽  
Number Of Customers

AbstractThis paper studies a single-sever queue with disasters and repairs, in which after each service completion the server may take a vacation with probabilityq(0≤q≤1), or begin to serve the next customer, if any, with probabilityp(= 1− q). The disaster only affects the system when the server is in operation, and once it occurs, all customers present are eliminated from the system. We obtain the stationary probability generating functions (PGFs) of the number of customers in the system by solving the balance equations of the system. Some performance measures such as the mean system length, the probability that the server is in different states, the rate at which disasters occur and the rate of initiations of busy period are determined. We also derive the sojourn time distribution and the mean sojourn time. In addition, some numerical examples are presented to show the effect of the parameters on the mean system length.

scholarly journals Analysis of a production system with a feedback buffer and a general dispatching time

2000 ◽  
Vol 5 (5) ◽  
pp. 421-439 ◽  
Author(s):  
Ho Woo Lee ◽  
Boo Yong Ahn
Keyword(s):  
Performance Measures ◽  
Production System ◽  
Sojourn Time ◽  
Operating Cost ◽  
Buffer Size ◽  
General Distribution ◽  
The Mean

In this paper, we analyze a production system with a finite feedback buffer and dispatching time. Parts enter a “main buffer” before they are processed. Processed parts leave the system with probability1−por are fed back to a “feedback buffer” with probabilityp. As soon as the feedback buffer becomes full, the parts in the feedback buffer are dispatched, all at once, to the main buffer by the server for reprocessing. The dispatching time follows a general distribution. Thus the server is engaged either in one of the following states: idle, processing, dispatching.We derive various performance measures such as the mean number of parts in each buffer, the mean system sojourn time and the dispatching rate. We also discuss the effects of the dispatching time on the performance measures. We finally derive the procedure to obtain the optimal buffer size that minimizes the overall operating cost.

Approximation of the Mean Queue Length of anM/G/cQueueing System

1995 ◽  
Vol 43 (1) ◽  
pp. 158-165 ◽  
Author(s):  
Bobby N. W. Ma ◽  
Jon W. Mark
Keyword(s):  
Queue Length ◽  
The Mean ◽  
Mean Queue Length
Sign in / Sign up

Export Citation Format

Share Document