scholarly journals Operational behavior of the MAP/G/1 queue under N-policy with a single vacation and set-up

2002 ◽  
Vol 15 (2) ◽  
pp. 151-180
Author(s):  
Ho Woo Lee ◽  
Boo Yong Ahn
Keyword(s):  
Performance Measures ◽  
Waiting Time ◽  
Generating Functions ◽  
Queue Length ◽  
Stieltjes Transform ◽  
Arbitrary Time ◽  
Single Vacation ◽  
Set Up ◽  
Computation Algorithms

This paper considers the MAP/G/1 queue under N-policy with a single vacation and set-up. We derive the vector generating functions of the queue length at an arbitrary time and at departures in decomposed forms. We also derive the Laplace-Stieltjes transform of the waiting time. Computation algorithms for mean performance measures 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.

scholarly journals On Mx/(G1G2)/1/G(BS)/Vs vacation queue with two types of general heterogeneous service

2005 ◽  
Vol 2005 (3) ◽  
pp. 123-135 ◽  
Author(s):  
Kailash C. Madan ◽  
Z. R. Al-Rawi ◽  
Amjad D. Al-Nasser
Keyword(s):  
Steady State ◽  
Performance Measures ◽  
Waiting Time ◽  
Busy Period ◽  
Single Server ◽  
Queue Size ◽  
Single Vacation ◽  
The Mean ◽  
Expected Waiting Time ◽  
Heterogeneous Service

We analyze a batch arrival queue with a single server providing two kinds of general heterogeneous service. Just before his service starts, a customer may choose one of the services and as soon as a service (of any kind) gets completed, the server may take a vacation or may continue staying in the system. The vacation times are assumed to be general and the server vacations are based on Bernoulli schedules under a single vacation policy. We obtain explicit queue size distribution at a random epoch as well as at a departure epoch and also the mean busy period of the server under the steady state. In addition, some important performance measures such as the expected queue size and the expected waiting time of a customer are obtained. Further, some interesting particular cases are also discussed.

scholarly journals A Analysis on Queuing System with Setup Time in Revamp Process

2019 ◽  
Vol 9 (1S4) ◽  
pp. 967-969
Keyword(s):  
Performance Measures ◽  
Generating Functions ◽  
Setup Time ◽  
Repair Process ◽  
The Other ◽  
Queuing Model ◽  
Second Stage ◽  
Time Stage ◽  
Set Up ◽  
Two Stages

In this paper, we study about a M/G/1 Queuing model with single stage of service. Service interrupts during the time of service. The server does not get into the repair process immediately. It gets into a Set up time stage for the prior work to be done. On completion of set up stage service, the server will get into the repair process consisting of two stages, in which first stage is compulsory and the second stage of service is optional. For the model defined, we get the steady state results in closed form in terms of the probability generating functions and all the other execution performance measures of the model defined.

scholarly journals M/M/1 Model with Unreliable Service

2017 ◽  
Vol 7 (1) ◽  
pp. 125
Author(s):  
Joshua Patterson, Andrzej Korzeniowski
Keyword(s):  
Markov Chain ◽  
Waiting Time ◽  
Queue Length ◽  
Sufficient Conditions ◽  
Probability Generating Function ◽  
Embedded Markov Chain ◽  
Queueing Models ◽  
Stieltjes Transform ◽  
Stationary Queue Length ◽  
Stationary Waiting Time

We define a new term ”unreliable service” and construct the corresponding embedded Markov Chain to an M/M/1 queue with so defined protocol. Sufficient conditions for positive recurrence and closed form of stationary distribution are provided. Furthermore, we compute the probability generating function of the stationary queue length and Laplace-Stieltjes transform of the stationary waiting time. In the course of the analysis an interesting decomposition of both the queue length and waiting time has emerged. A number of queueing models can be recovered from our work by taking limits of certain parameters.

Some general results for many server queues

1973 ◽  
Vol 5 (01) ◽  
pp. 153-169 ◽  
Author(s):  
J. H. A. De Smit
Keyword(s):  
Integral Equations ◽  
Waiting Time ◽  
Queue Length ◽  
Joint Distribution ◽  
Waiting Times ◽  
Virtual Waiting Time ◽  
Server Queue ◽  
Many Server Queues ◽  
The Many ◽  
Busy Cycle

Pollaczek's theory for the many server queue is generalized and extended. Pollaczek (1961) found the distribution of the actual waiting times in the model G/G/s as a solution of a set of integral equations. We give a somewhat more general set of integral equations from which the joint distribution of the actual waiting time and some other random variables may be found. With this joint distribution we can obtain distributions of a number of characteristic quantities, such as the virtual waiting time, the queue length, the number of busy servers, the busy period and the busy cycle. For a wide class of many server queues the formal expressions may lead to explicit results.

Some analyses on the control of queues using level crossings of regenerative processes

1980 ◽  
Vol 17 (3) ◽  
pp. 814-821 ◽  
Author(s):  
J. G. Shanthikumar
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Queueing Systems ◽  
Density Functions ◽  
Waiting Time Distribution ◽  
Regenerative Processes ◽  
Stieltjes Transform ◽  
Expected Number ◽  
Control Of Queues ◽  
Special Case

Some properties of the number of up- and downcrossings over level u, in a special case of regenerative processes are discussed. Two basic relations between the density functions and the expected number of upcrossings of this process are derived. Using these reults, two examples of controlled M/G/1 queueing systems are solved. Simple relations are derived for the waiting time distribution conditioned on the phase of control encountered by an arriving customer. The Laplace-Stieltjes transform of the distribution function of the waiting time of an arbitrary customer is also derived for each of these two examples.

scholarly journals Decomposition of M/M/1 With Unreliable Service and a Working Vacation

2020 ◽  
Vol 9 (1) ◽  
pp. 63
Author(s):  
Joshua Patterson ◽  
Andrzej Korzeniowski
Keyword(s):  
Closed Form ◽  
Failure Rate ◽  
Waiting Time ◽  
Queue Length ◽  
Service Failure ◽  
Working Vacation ◽  
Stationary Queue Length ◽  
Stationary Waiting Time ◽  
Relationship Of ◽  
The Relationship

We use the stationary distribution for the M/M/1 with Unreliable Service and aWorking Vacation (M/M/1/US/WV) given explicitly in (Patterson & Korzeniowski, 2019) to find a decomposition of the stationary queue length N. By applying the distributional form of Little's Law the Laplace-tieltjes Transform of the stationary customer waiting time W is derived. The closed form of the expected value and variance for both N and W is found and the relationship of the expected stationary waiting time as a function of the service failure rate is determined.

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 and queue length distributions for go-back-N and selective-repeat ARQ protocols

1993 ◽  
Vol 41 (11) ◽  
pp. 1687-1693 ◽  
Author(s):  
M. Yoshimoto ◽  
T. Takine ◽  
Y. Takahashi ◽  
T. Hasegawa
Keyword(s):  
Waiting Time ◽  
Queue Length

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.

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