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.

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.

APPROXIMATION OF PH/PH/c RETRIAL QUEUE WITH PH-RETRIAL TIME

2014 ◽  
Vol 31 (02) ◽  
pp. 1440010 ◽  
Author(s):  
YANG WOO SHIN ◽  
DUG HEE MOON
Keyword(s):  
Time Distribution ◽  
Retrial Queue ◽  
Retrial Queues ◽  
General Distribution ◽  
Phase Type Distribution ◽  
Type Distribution ◽  
Phase Type ◽  
Sojourn Time Distribution ◽  
The Mean ◽  
Number Of Customers

We consider the PH/PH/c retrial queues with PH-retrial time. Approximation formulae for the distribution of the number of customers in service facility, sojourn time distribution and the mean number of customers in orbit are presented. We provide an approximation for GI/G/c retrial queue with general retrial time by approximating the general distribution with phase type distribution. Some numerical results are presented.

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.

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.

Reducing Delay in Retrial Queues by Simultaneously Differentiating Service and Retrial Rates

2020 ◽  
Vol 68 (6) ◽  
pp. 1648-1667
Author(s):  
Jinting Wang ◽  
Zhongbin Wang ◽  
Yunan Liu
Keyword(s):  
Mobile Networks ◽  
Heavy Traffic ◽  
Inventory Systems ◽  
Service Systems ◽  
Retrial Queues ◽  
Complex Nature ◽  
Service Capacity ◽  
Asymptotic Reduction ◽  
Additional Service ◽  
The Impact

Customer retrials commonly occur in many service systems, such as healthcare, call centers, mobile networks, computer systems, and inventory systems. However, because of their complex nature, retrial queues are often more difficult to analyze than queues without retrials. In “Reducing Delay in Retrial Queues by Simultaneously Differentiating Service and Retrial Rates”, J. Wang, Z. Wang, and Y. Liu develop a service grade differentiation policy for queueing models with customer retrials. They show that the average waiting time can be reduced through strategically allocating the rates of service and retrial times without needing additional service capacity. Counter to the intuition that higher service variability usually yields a larger delay, the authors show that the benefits of this simultaneous service-and-retrial differentiation (SSRD) policy outweigh the impact of the increased service variability. To validate the effectiveness of the new SSRD policy, the authors provide (i) conditions under which SSRD is more beneficial, (ii) closed-form expressions of the optimal policy, (iii) asymptotic reduction of customer delays when the system is in heavy traffic, and (iv) insightful observations/discussions and numerical results.

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 Performance of an M/M/1 Retrial Queue with Working Vacation Interruption and Classical Retrial Policy

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Tao Li ◽  
Liyuan Zhang ◽  
Shan Gao
Keyword(s):  
Generating Functions ◽  
System Performance ◽  
Cost Optimization ◽  
Retrial Queue ◽  
Stable Condition ◽  
Working Vacation ◽  
Optimization Analysis ◽  
Vacation Interruption ◽  
Number Of Customers ◽  
Vacation Period

An M/M/1 retrial queue with working vacation interruption is considered. Upon the arrival of a customer, if the server is busy, it would join the orbit of infinite size. The customers in the orbit will try for service one by one when the server is idle under the classical retrial policy with retrial ratenα, wherenis the size of the orbit. During a working vacation period, if there are customers in the system at a service completion instant, the vacation will be interrupted. Under the stable condition, the probability generating functions of the number of customers in the orbit are obtained. Various system performance measures are also developed. Finally, some numerical examples and cost optimization analysis are presented.

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.

INFINITE-SERVER QUEUES WITH SYSTEM'S ADDITIONAL TASKS AND IMPATIENT CUSTOMERS

2008 ◽  
Vol 22 (4) ◽  
pp. 477-493 ◽  
Author(s):  
Eitan Altman ◽  
Uri Yechiali
Keyword(s):  
Quality Of Service ◽  
Generating Functions ◽  
Busy Period ◽  
Impatient Customers ◽  
Partial Probability ◽  
The Mean ◽  
Infinite Server Queues ◽  
Mean Cycle ◽  
Number Of Customers

A system is operating as an M/M/∞ queue. However, when it becomes empty, it is assigned to perform another task, the duration U of which is random. Customers arriving while the system is unavailable for service (i.e., occupied with a U-task) become impatient: Each individual activates an “impatience timer” having random duration T such that if the system does not become available by the time the timer expires, the customer leaves the system never to return. When the system completes a U-task and there are waiting customers, each one is taken immediately into service. We analyze both multiple and single U-task scenarios and consider both exponentially and generally distributed task and impatience times. We derive the (partial) probability generating functions of the number of customers present when the system is occupied with a U-task as well as when it acts as an M/M/∞ queue and we obtain explicit expressions for the corresponding mean queue sizes. We further calculate the mean length of a busy period, the mean cycle time, and the quality of service measure: proportion of customers being served.

The sojourn-time distribution in the M/G/1 queue by processor sharing

1984 ◽  
Vol 21 (02) ◽  
pp. 360-378
Author(s):  
Teunis J. Ott
Keyword(s):  
Generating Functions ◽  
Sojourn Time ◽  
Time Distribution ◽  
Joint Distribution ◽  
Processor Sharing ◽  
Explicit Formulas ◽  
Sojourn Time Distribution ◽  
Number Of Customers ◽  
Stieltjes Transforms

This paper gives, in the form of Laplace–Stieltjes transforms and generating functions, the joint distribution of the sojourn time and the number of customers in the system at departure for customers in the general M/G/1 queue with processor sharing (M/G/1/PS). Explicit formulas are given for a number of conditional and unconditional moments, including the variance of the sojourn time of an ‘arbitrary' customer.

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