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.

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.

Waiting Time in M/G/1 Queues with Impolite Arrival Disciplines

1995 ◽  
Vol 9 (2) ◽  
pp. 255-267 ◽  
Author(s):  
Süleyman Òzekici ◽  
Jingwen Li ◽  
Fee Seng Chou
Keyword(s):  
Performance Measures ◽  
Waiting Time ◽  
Time Distribution ◽  
Iterative Procedure ◽  
Queueing System ◽  
Waiting Time Distribution ◽  
Special Model ◽  
Average Waiting Time ◽  
Random Position ◽  
Number Of Customers

We consider a queueing system where arriving customers join the queue at some random position. This constitutes an impolite arrival discipline because customers do not necessarily go to the end of the line upon arrival. Although mean performance measures like the average waiting time and average number of customers in the queue are the same for all such disciplines, we show that the variance of the waiting time increases as the arrival discipline becomes more impolite, in the sense that a customer is more likely to choose a position closer to the server. For the M/G/1 model, we also provide an iterative procedure for computing the moments of the waiting time distribution. Explicit computational formulas are derived for an interesting special model where a customer joins the queue either at the head or at the end of the line.

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.

scholarly journals M/G/1/K system with push-out scheme under vacation policy

1996 ◽  
Vol 9 (2) ◽  
pp. 143-157 ◽  
Author(s):  
Shoji Kasahara ◽  
Hideaki Takagi ◽  
Yutaka Takahashi ◽  
Toshiharu Hasegawa
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Waiting Times ◽  
Numerical Results ◽  
The Other ◽  
Waiting Time Distribution ◽  
Stieltjes Transform ◽  
Multiple Vacations ◽  
Push Out

We consider an M/G/1/K system with push-out scheme and multiple vacations. This model is particularly important in situations where it is essential to provide short waiting times to messages which are selected for service. We analyze the behavior of two types of messages: one that succeeds in transmission and the other that fails. We derive the Laplace-Stieltjes transform of the waiting time distribution for the message which is eventually served. Finally, we show some numerical results including the comparisons between the push-out and the ordinary blocking models.

GeoX1, GeoX2/D/c HOL Priority Queueing System with Random Order Selection within Each Priority Class

1998 ◽  
Vol 12 (1) ◽  
pp. 125-139 ◽  
Author(s):  
Bong Dae Choi ◽  
Yutae Lee ◽  
Doo Il Choi
Keyword(s):  
Queue Length ◽  
Time Distribution ◽  
Queueing System ◽  
Length Distribution ◽  
Random Order ◽  
Order Selection ◽  
Waiting Time Distribution ◽  
Queue Length Distribution ◽  
Packet Switch ◽  
Priority Queueing

We model the virtual contention queue in an ATM nonblocking packet switch with capacity c and input queues by a Geox1, Geox2/D/c Head-of-Line priority queueing system with Random Order Selection within each class and find the joint queue length distribution and the waiting time distribution for each class.

A note on waiting times in systems with queues in parallel

1987 ◽  
Vol 24 (2) ◽  
pp. 540-546 ◽  
Author(s):  
J. P. C. Blanc
Keyword(s):  
Queue Length ◽  
Time Distribution ◽  
Waiting Times ◽  
Numerical Data ◽  
Series Expansions ◽  
Waiting Time Distribution ◽  
Power Series Expansions ◽  
The Mean ◽  
Waiting Line ◽  
Queues In Parallel

Numerical data are presented concerning the mean and the standard deviation of the waiting-time distribution for multiserver systems with queues in parallel, in which customers choose one of the shortest queues upon arrival. Moreover, a new numerical method is outlined for calculating state probabilities and moments of queue-length distributions. This method is based on power series expansions and recursion. It is applicable to many systems with more than one waiting line.

scholarly journals Single server queues with modified service mechanisms

1962 ◽  
Vol 2 (4) ◽  
pp. 499-507 ◽  
Author(s):  
G. F. Yeo
Keyword(s):  
Time Distribution ◽  
Busy Period ◽  
Queueing System ◽  
Probability Generating Function ◽  
Single Server ◽  
Waiting Time Distribution ◽  
Period Distribution ◽  
Time Dependent Problem ◽  
Stationary Waiting Time ◽  
Special Case

SummaryThis paper considers a generalisation of the queueing system M/G/I, where customers arriving at empty and non-empty queues have different service time distributions. The characteristic function (c.f.) of the stationary waiting time distribution and the probability generating function (p.g.f.) of the queue size are obtained. The busy period distribution is found; the results are generalised to an Erlangian inter-arrival distribution; the time-dependent problem is considered, and finally a special case of server absenteeism is discussed.

On the waiting time distribution in a generalized queueing system with uniformly bounded sojourn times

1972 ◽  
Vol 9 (3) ◽  
pp. 642-649 ◽  
Author(s):  
Jacqueline Loris-Teghem
Keyword(s):  
Generating Function ◽  
Waiting Time ◽  
Service Time ◽  
Time Distribution ◽  
Queueing System ◽  
Waiting Time Distribution ◽  
Stieltjes Transform ◽  
Sojourn Times ◽  
Stieltjes Transforms ◽  
Uniformly Bounded

A generalized queueing system with (N + 2) types of triplets (delay, service time, probability of joining the queue) and with uniformly bounded sojourn times is considered. An expression for the generating function of the Laplace-Stieltjes transforms of the waiting time distributions is derived analytically, in a case where some of the random variables defining the model have a rational Laplace-Stieltjes transform.The standard Kl/Km/1 queueing system with uniformly bounded sojourn times is considered in particular.

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.

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