JOINT STATIONARY DISTRIBUTION OF THE NUMBER OF CUSTOMERS IN THE SYSTEM AND REORDERING BUFFER IN THE MULTISERVER REORDERING QUEUE

2014 ◽  
Keyword(s):  
Stationary Distribution ◽  
Joint Stationary Distribution ◽  
Number Of Customers

The Mk/G/∞ batch arrival queue by heterogeneous dependent demands

1994 ◽  
Vol 31 (03) ◽  
pp. 841-846
Author(s):  
Gennadi Falin
Keyword(s):  
Linear Function ◽  
Stationary Distribution ◽  
Random Variables ◽  
Dimensional Vector ◽  
Transient Solution ◽  
General Service Times ◽  
Joint Stationary Distribution ◽  
Number Of Customers ◽  
General Service ◽  
Server System

Choi and Park [2] derived an expression for the joint stationary distribution of the number of customers of k types who arrive in batches at an infinite-server system of M/M/∞ type. We propose another method of solving this problem and extend the result to the case of general service times (not necessarily independent). We also get a transient solution. Our main result states that the k- dimensional vector of the number of customers of k types in the system is a certain linear function of a (2 k – 1)-dimensional vector whose coordinates are independent Poisson random variables.

The Mk/G/∞ batch arrival queue by heterogeneous dependent demands

1994 ◽  
Vol 31 (3) ◽  
pp. 841-846 ◽  
Author(s):  
Gennadi Falin
Keyword(s):  
Linear Function ◽  
Stationary Distribution ◽  
Random Variables ◽  
Dimensional Vector ◽  
Transient Solution ◽  
General Service Times ◽  
Joint Stationary Distribution ◽  
Number Of Customers ◽  
General Service ◽  
Server System

Choi and Park [2] derived an expression for the joint stationary distribution of the number of customers of k types who arrive in batches at an infinite-server system of M/M/∞ type. We propose another method of solving this problem and extend the result to the case of general service times (not necessarily independent). We also get a transient solution. Our main result states that the k- dimensional vector of the number of customers of k types in the system is a certain linear function of a (2k – 1)-dimensional vector whose coordinates are independent Poisson random variables.

scholarly journals Stationary solution to the fluid queue fed by an M/M/1 queue

2002 ◽  
Vol 39 (2) ◽  
pp. 359-369 ◽  
Author(s):  
N. Barbot ◽  
B. Sericola
Keyword(s):  
Stationary Solution ◽  
Stationary Distribution ◽  
Generating Functions ◽  
The State ◽  
Finite Capacity ◽  
Fluid Queue ◽  
Analytic Expression ◽  
Joint Stationary Distribution

We consider an infinite-capacity buffer receiving fluid at a rate depending on the state of an M/M/1 queue. We obtain a new analytic expression for the joint stationary distribution of the buffer level and the state of the M/M/1 queue. This expression is obtained by the use of generating functions which are explicitly inverted. The case of a finite capacity fluid queue is also considered.

Fluid Model Driven by an M/M/1 Queue with Set-Up and Close-Down Period

2014 ◽  
Vol 513-517 ◽  
pp. 3377-3380
Author(s):  
Fu Wei Wang ◽  
Bing Wei Mao
Keyword(s):  
Stationary Distribution ◽  
Solution Method ◽  
Fluid Model ◽  
Model Driven ◽  
Buffer Content ◽  
The Laplace Transform ◽  
Matrix Geometric Solution ◽  
The Mean ◽  
Joint Stationary Distribution ◽  
Set Up

The fluid model driven by an M/M/1 queue with set-up and close-down period is studied. The Laplace transform of the joint stationary distribution of the fluid model is of matrix geometric structure. With matrix geometric solution method, the Laplace-Stieltjes transformation of the stationary distribution of the buffer content is obtained, as well as the mean buffer content. Finally, with some numerical examples, the effect of the parameters on mean buffer content is presented.

scholarly journals Multiserver Queue with Guard Channel for Priority and Retrial Customers

2016 ◽  
Vol 2016 ◽  
pp. 1-23 ◽  
Author(s):  
Kazuki Kajiwara ◽  
Tuan Phung-Duc
Keyword(s):  
Stationary Distribution ◽  
Queueing System ◽  
Taylor Series Expansion ◽  
Retrial Queueing ◽  
Guard Channel ◽  
Guard Channels ◽  
Qbd Process ◽  
Joint Stationary Distribution ◽  
Idle Channel ◽  
Asymptotic Upper Bound

This paper considers a retrial queueing model where a group of guard channels is reserved for priority and retrial customers. Priority and normal customers arrive at the system according to two distinct Poisson processes. Priority customers are accepted if there is an idle channel upon arrival while normal customers are accepted if and only if the number of idle channels is larger than the number of guard channels. Blocked customers (priority or normal) join a virtual orbit and repeat their attempts in a later time. Customers from the orbit (retrial customers) are accepted if there is an idle channel available upon arrival. We formulate the queueing system using a level dependent quasi-birth-and-death (QBD) process. We obtain a Taylor series expansion for the nonzero elements of the rate matrices of the level dependent QBD process. Using the expansion results, we obtain an asymptotic upper bound for the joint stationary distribution of the number of busy channels and that of customers in the orbit. Furthermore, we develop an efficient numerical algorithm to calculate the joint stationary distribution.

scholarly journals Stationary solution to the fluid queue fed by an M/M/1 queue

2002 ◽  
Vol 39 (02) ◽  
pp. 359-369 ◽  
Author(s):  
N. Barbot ◽  
B. Sericola
Keyword(s):  
Stationary Solution ◽  
Stationary Distribution ◽  
Generating Functions ◽  
The State ◽  
Finite Capacity ◽  
Fluid Queue ◽  
Analytic Expression ◽  
Joint Stationary Distribution

We consider an infinite-capacity buffer receiving fluid at a rate depending on the state of an M/M/1 queue. We obtain a new analytic expression for the joint stationary distribution of the buffer level and the state of the M/M/1 queue. This expression is obtained by the use of generating functions which are explicitly inverted. The case of a finite capacity fluid queue is also considered.

MULTISERVER RETRIAL QUEUES WITH TWO TYPES OF NONPERSISTENT CUSTOMERS

2014 ◽  
Vol 31 (02) ◽  
pp. 1440009 ◽  
Author(s):  
TUAN PHUNG-DUC
Keyword(s):  
Fixed Point ◽  
Generating Functions ◽  
Service Systems ◽  
Retrial Queues ◽  
Approximation Model ◽  
System Behavior ◽  
Stable Algorithm ◽  
Qbd Process ◽  
Joint Stationary Distribution ◽  
Number Of Customers

We consider M/M/c/K(K ≥ c ≥ 1) retrial queues with two types of nonpersistent customers, which are motivated from modeling of service systems such as call centers. Arriving customers that see the system fully occupied either join the orbit or abandon receiving service forever. After an exponentially distributed time in the orbit, each customer either abandons the system forever or retries to occupy a server again. For the case of K = c = 1, we present an analytical solution for the generating functions in terms of confluent hypegeometric functions. In the general case, the number of customers in the system and that in the orbit form a level-dependent quasi-birth-and-death (QBD) process whose structure is sparse. Based on this sparse structure, we develop a numerically stable algorithm to compute the joint stationary distribution. We show that the computational complexity of the algorithm is linear to the capacity of the queue. Furthermore, we present a simple fixed point approximation model for the case where the algorithm is time consuming. Numerical results show various insights into the system behavior.

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