scholarly journals On the Discrete-TimeGeoX/G/1Queues underN-Policy with Single and Multiple Vacations

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Sung J. Kim ◽  
Nam K. Kim ◽  
Hyun-Min Park ◽  
Kyung Chul Chae ◽  
Dae-Eun Lim
Keyword(s):  
Queue Length ◽  
Queueing System ◽  
Threshold Value ◽  
Stochastic Decomposition ◽  
Idle Period ◽  
Multiple Vacations ◽  
Vacation Models ◽  
Single Vacation ◽  
Special Cases ◽  
Stationary Queue Length

We consider the discrete-timeGeoX/G/1queue underN-policy with single and multiple vacations. In this queueing system, the server takes multiple vacations and a single vacation whenever the system becomes empty and begins to serve customers only if the queue length is at least a predetermined threshold valueN. Using the well-known property of stochastic decomposition, we derive the stationary queue-length distributions for both vacation models in a simple and unified manner. In addition, we derive their busy as well as idle-period distributions. Some classical vacation models are considered as special cases.

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.

Transient Analysis of a Two-Heterogeneous Severs Queue with Impatient Behaviour and Multiple Vacations

2018 ◽  
Vol 6 (1) ◽  
pp. 69-84
Author(s):  
Jia Xu ◽  
Liwei Liu ◽  
Taozeng Zhu
Keyword(s):  
Generating Functions ◽  
Transient Analysis ◽  
Bessel Functions ◽  
Queueing System ◽  
System Size ◽  
Heterogeneous Servers ◽  
Modified Bessel Functions ◽  
Multiple Vacations ◽  
Special Cases ◽  
Mean And Variance

AbstractWe consider anM/M/2 queueing system with two-heterogeneous servers and multiple vacations. Customers arrive according to a Poisson process. However, customers become impatient when the system is on vacation. We obtain explicit expressions for the time dependent probabilities, mean and variance of the system size at timetby employing probability generating functions, continued fractions and properties of the modified Bessel functions. Finally, two special cases are provided.

scholarly journals The Geo/Geo/1+1 Queueing System with Negative Customers

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Zhanyou Ma ◽  
Yalin Guo ◽  
Pengcheng Wang ◽  
Yumei Hou
Keyword(s):  
Performance Measures ◽  
Waiting Time ◽  
System Performance ◽  
Queue Length ◽  
Solution Method ◽  
Queueing System ◽  
Negative Customers ◽  
Idle Period ◽  
Matrix Geometric Solution ◽  
Qbd Process

We study a Geo/Geo/1+1 queueing system with geometrical arrivals of both positive and negative customers in which killing strategies considered are removal of customers at the head (RCH) and removal of customers at the end (RCE). Using quasi-birth-death (QBD) process and matrix-geometric solution method, we obtain the stationary distribution of the queue length, the average waiting time of a new arrival customer, and the probabilities of servers in busy or idle period, respectively. Finally, we analyze the effect of some related parameters on the system performance measures.

A note on the MX/GY/1, K bulk queueing system

1973 ◽  
Vol 10 (04) ◽  
pp. 901-906
Author(s):  
Tapan P. Bagchi ◽  
J. G. C. Templeton
Keyword(s):  
Queue Length ◽  
Queueing System ◽  
Present Note ◽  
Stationary Queue Length ◽  
Number Of Customers

Cohen (1969) has studied the transient and stationary queue length distributions for the M/G/1, K queue, with a fixed maximum number of customers, K, in the system at any time. The present note applies Cohen's method to generalize his results to the MX/GY/1, K queue.

ANALYSIS OF A BULK QUEUE WITH FAST AND SLOW SERVICE RATES AND MULTIPLE VACATIONS

2005 ◽  
Vol 22 (02) ◽  
pp. 239-260 ◽  
Author(s):  
R. ARUMUGANATHAN ◽  
K. S. RAMASWAMI
Keyword(s):  
Performance Measures ◽  
Queue Length ◽  
Queueing System ◽  
Cost Model ◽  
Probability Generating Function ◽  
Service Rate ◽  
Multiple Vacations ◽  
Bulk Queue ◽  
Service Rates ◽  
Number Of Customers

We analyze a Mx/G(a,b)/1 queueing system with fast and slow service rates and multiple vacations. The server does the service with a faster rate or a slower rate based on the queue length. At a service completion epoch (or) at a vacation completion epoch if the number of customers waiting in the queue is greater than or equal to N (N > b), then the service is rendered at a faster rate, otherwise with a slower service rate. After finishing a service, if the queue length is less than 'a' the server leaves for a vacation of random length. When he returns from the vacation, if the queue length is still less than 'a' he leaves for another vacation and so on until he finally finds atleast 'a' customers waiting for service. After a service (or) a vacation, if the server finds atleast 'a' customers waiting for service say ξ, then he serves a batch of min (ξ, b) customers, where b ≥ a. We derive the probability generating function of the queue size at an arbitrary time. Various performance measures are obtained. A cost model is discussed with a numerical solution.

The analysis of discrete time Geom/Geom/1 queue with single working vacation and multiple vacations (Geom/Geom/1/SWV+MV)

2018 ◽  
Vol 52 (1) ◽  
pp. 95-117 ◽  
Author(s):  
Qingqing Ye ◽  
Liwei Liu
Keyword(s):  
Discrete Time ◽  
System Size ◽  
Stochastic Decomposition ◽  
Model Parameters ◽  
Working Vacation ◽  
Two Phase ◽  
Multiple Vacations ◽  
Special Cases ◽  
The Matrix ◽  
Matrix Geometric Solution

In this article, we consider a discrete-time Geom/Geom/1 queue with two phase vacation policy that comprises single working vacation and multiple vacations, denoted by Geom/Geom/1/SWV+MV. For this model, we first derive the explicit expression for the stationary system size by the matrix-geometric solution method. Next, we obtain the stochastic decomposition structures of system size and the sojourn time of an arbitrary customer in steady state. Moreover, the regular busy period and busy cycle are analyzed by limiting theorem of alternative renewal process. Besides, some special cases are presented and the relationship between the Geom/Geom/1/SWV+MV queue and its continuous time counterpart is investigated. Finally, we perform several experiments to illustrate the effect of model parameters on some performance measures.

scholarly journals Repairable Queue with Non-exponential Interarrival Time and Variable Breakdown Rates

2018 ◽  
Vol 7 (2.15) ◽  
pp. 76
Author(s):  
Koh Siew Khew ◽  
Chin Ching Herny ◽  
Tan Yi Fei ◽  
Pooi Ah Hin ◽  
Goh Yong Kheng ◽  
...  
Keyword(s):  
Queue Length ◽  
Length Distribution ◽  
Cumulative Distribution ◽  
Interarrival Time ◽  
Single Server ◽  
Idle Period ◽  
Queue Length Distribution ◽  
Breakdown Rates ◽  
Stationary Queue Length ◽  
Stationary Queue Length Distribution

This paper considers a single server queue in which the service time is exponentially distributed and the service station may breakdown according to a Poisson process with the rates γ and γ' in busy period and idle period respectively. Repair will be performed immediately following a breakdown. The repair time is assumed to have an exponential distribution. Let g(t) and G(t) be the probability density function and the cumulative distribution function of the interarrival time respectively. When t tends to infinity, the rate of g(t)/[1 – G(t)] will tend to a constant. A set of equations will be derived for the probabilities of the queue length and the states of the arrival, repair and service processes when the queue is in a stationary state. By solving these equations, numerical results for the stationary queue length distribution can be obtained. 

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

1994 ◽  
Vol 31 (2) ◽  
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.

scholarly journals An alternative approach in finding the stationary queue length distribution of a queueing system with negative customers

2021 ◽  
Vol 36 ◽  
pp. 04001
Author(s):  
Siew Khew Koh ◽  
Ching Herny Chin ◽  
Yi Fei Tan ◽  
Tan Ching Ng
Keyword(s):  
Queue Length ◽  
Queueing System ◽  
Length Distribution ◽  
Interarrival Time ◽  
Negative Customers ◽  
Queue Length Distribution ◽  
Alternative Approach ◽  
Stationary Queue Length ◽  
Stationary Queue Length Distribution ◽  
Stationary Probabilities

A single-server queueing system with negative customers is considered in this paper. One positive customer will be removed from the head of the queue if any negative customer is present. The distribution of the interarrival time for the positive customer is assumed to have a rate that tends to a constant as time t tends to infinity. An alternative approach will be proposed to derive a set of equations to find the stationary probabilities. The stationary probabilities will then be used to find the stationary queue length distribution. Numerical examples will be presented and compared to the results found using the analytical method and simulation procedure. The advantage of using the proposed alternative approach will be discussed in this paper.

Sign in / Sign up

Export Citation Format

Share Document