scholarly journals Analysis of polling models with a self-ruling server

2019 ◽  
Vol 94 (1-2) ◽  
pp. 77-107 ◽  
Author(s):  
Jan-Kees van Ommeren ◽  
Ahmad Al Hanbali ◽  
Richard J. Boucherie
Keyword(s):  
Probability Generating Function ◽  
Joint Probability ◽  
Polling Systems ◽  
Single Server ◽  
Batch Arrivals ◽  
Exponential Time ◽  
Polling Models ◽  
The Laplace Transform ◽  
Expected Sojourn Time ◽  
Number Of Customers

AbstractPolling systems are systems consisting of multiple queues served by a single server. In this paper, we analyze polling systems with a server that is self-ruling, i.e., the server can decide to leave a queue, independent of the queue length and the number of served customers, or stay longer at a queue even if there is no customer waiting in the queue. The server decides during a service whether this is the last service of the visit and to leave the queue afterward, or it is a regular service followed, possibly, by other services. The characteristics of the last service may be different from the other services. For these polling systems, we derive a relation between the joint probability generating functions of the number of customers at the start of a server visit and, respectively, at the end of a server visit. We use these key relations to derive the joint probability generating function of the number of customers and the Laplace transform of the workload in the queues at an arbitrary time. Our analysis in this paper is a generalization of several models including the exponential time-limited model with preemptive-repeat-random service, the exponential time-limited model with non-preemptive service, the gated time-limited model, the Bernoulli time-limited model, the 1-limited discipline, the binomial gated discipline, and the binomial exhaustive discipline. Finally, we apply our results on an example of a new polling discipline, called the 1 + 1 self-ruling server, with Poisson batch arrivals. For this example, we compute numerically the expected sojourn time of an arbitrary customer in the queues.

Transient analysis for exponential time-limited polling models under the preemptive repeat random policy

2020 ◽  
Vol 52 (1) ◽  
pp. 32-60
Author(s):  
Roland De Haan ◽  
Ahmad Al Hanbali ◽  
Richard J. Boucherie ◽  
Jan-Kees Van Ommeren
Keyword(s):  
Queue Length ◽  
Transient Analysis ◽  
Busy Period ◽  
Queueing Systems ◽  
Polling Systems ◽  
Single Server ◽  
Direct Relation ◽  
Batch Arrivals ◽  
Exponential Time ◽  
Polling Models

AbstractPolling systems are queueing systems consisting of multiple queues served by a single server. In this paper we analyze two types of preemptive time-limited polling systems, the so-called pure and exhaustive time-limited disciplines. In particular, we derive a direct relation for the evolution of the joint queue length during the course of a server visit. The analysis of the pure time-limited discipline builds on and extends several known results for the transient analysis of an M/G/1 queue. For the analysis of the exhaustive discipline we derive several new results for the transient analysis of the M/G/1 queue during a busy period. The final expressions for both types of polling systems that we obtain generalize previous results by incorporating customer routeing, generalized service times, batch arrivals, and Markovian polling of the server.

Busy-period analysis of a correlated queue with exponential demand and service

1987 ◽  
Vol 24 (2) ◽  
pp. 476-485 ◽  
Author(s):  
Christos Langaris
Keyword(s):  
Probability Density Function ◽  
Busy Period ◽  
Joint Probability ◽  
Closed Form Expression ◽  
Single Server ◽  
Form Expression ◽  
Period Analysis ◽  
The Laplace Transform ◽  
Number Of Customers ◽  
Period Duration

In this paper we investigate the server's busy period in a single-server queueing situation in which the interarrival interval T preceding the arrival of a customer and his service time S are assumed correlated. A closed-form expression is obtained for the Laplace transform bn(z) of the joint probability and probability density function of the busy period duration and the number of customers served in it. Some numerical values are given showing the effect of correlation between T and S.

Busy-period analysis of a correlated queue with exponential demand and service

1987 ◽  
Vol 24 (02) ◽  
pp. 476-485 ◽  
Author(s):  
Christos Langaris
Keyword(s):  
Probability Density Function ◽  
Busy Period ◽  
Joint Probability ◽  
Closed Form Expression ◽  
Single Server ◽  
Form Expression ◽  
Period Analysis ◽  
The Laplace Transform ◽  
Number Of Customers ◽  
Period Duration

In this paper we investigate the server's busy period in a single-server queueing situation in which the interarrival interval T preceding the arrival of a customer and his service time S are assumed correlated. A closed-form expression is obtained for the Laplace transform bn (z) of the joint probability and probability density function of the busy period duration and the number of customers served in it. Some numerical values are given showing the effect of correlation between T and S.

scholarly journals A TWO-CLASS RETRIAL SYSTEM WITH COUPLED ORBIT QUEUES

2017 ◽  
Vol 31 (2) ◽  
pp. 139-179 ◽  
Author(s):  
Ioannis Dimitriou
Keyword(s):  
Performance Metrics ◽  
Kernel Method ◽  
Length Distribution ◽  
Probability Generating Function ◽  
Joint Probability ◽  
Joint Probability Distribution ◽  
Single Server ◽  
Method Performance ◽  
Service Station ◽  
Queue Length Distribution

We consider a single server system accepting two types of retrial customers, which arrive according to two independent Poisson streams. The service station can handle at most one customer, and in case of blocking, typeicustomer,i=1, 2, is routed to a separate typeiorbit queue of infinite capacity. Customers from the orbits try to access the server according to the constant retrial policy. We consider coupled orbit queues, and thus, when both orbit queues are non-empty, the orbit queueitries to re-dispatch a blocked customer of typeito the main service station after an exponentially distributed time with rate μi. If an orbit queue empties, the other orbit queue changes its re-dispatch rate from μito$\mu_{i}^{\ast}$. We consider both exponential and arbitrary distributed service requirements, and show that the probability generating function of the joint stationary orbit queue length distribution can be determined using the theory of Riemann (–Hilbert) boundary value problems. For exponential service requirements, we also investigate the exact tail asymptotic behavior of the stationary joint probability distribution of the two orbits with either an idle or a busy server by using the kernel method. Performance metrics are obtained, computational issues are discussed and a simple numerical example is presented.

Transient and steady-state analysis of a queuing system having customers’ impatience with threshold

2019 ◽  
Vol 53 (5) ◽  
pp. 1861-1876 ◽  
Author(s):  
Sapana Sharma ◽  
Rakesh Kumar ◽  
Sherif Ibrahim Ammar
Keyword(s):  
Steady State ◽  
Probability Generating Function ◽  
Threshold Value ◽  
Steady State Solution ◽  
Function Technique ◽  
Single Server ◽  
Queuing Model ◽  
Queuing Models ◽  
Special Cases ◽  
Number Of Customers

In many practical queuing situations reneging and balking can only occur if the number of customers in the system is greater than a certain threshold value. Therefore, in this paper we study a single server Markovian queuing model having customers’ impatience (balking and reneging) with threshold, and retention of reneging customers. The transient analysis of the model is performed by using probability generating function technique. The expressions for the mean and variance of the number of customers in the system are obtained and a numerical example is also provided. Further the steady-state solution of the model is obtained. Finally, some important queuing models are derived as the special cases of this model.

scholarly journals M/G/∞ POLLING SYSTEMS WITH RANDOM VISIT TIMES

2007 ◽  
Vol 22 (1) ◽  
pp. 81-106 ◽  
Author(s):  
M. Vlasiou ◽  
U. Yechiali
Keyword(s):  
Service Time ◽  
Probability Generating Function ◽  
Expected Value ◽  
Polling Systems ◽  
Service Time Distribution ◽  
Polling System ◽  
Stieltjes Transform ◽  
Polling Model ◽  
Queue Lengths ◽  
Number Of Customers

We consider a polling system where a group of an infinite number of servers visits sequentially a set of queues. When visited, each queue is attended for a random time. Arrivals at each queue follow a Poisson process, and the service time of each individual customer is drawn from a general probability distribution function. Thus, each of the queues comprising the system is, in isolation, anM/G/∞-type queue. A job that is not completed during a visit will have a new service-time requirement sampled from the service-time distribution of the corresponding queue. To the best of our knowledge, this article is the first in which anM/G/∞-type polling system is analyzed. For this polling model, we derive the probability generating function and expected value of the queue lengths and the Laplace–Stieltjes transform and expected value of the sojourn time of a customer. Moreover, we identify the policy that maximizes the throughput of the system per cycle and conclude that under the Hamiltonian-tour approach, the optimal visiting order isindependentof the number of customers present at the various queues at the start of the cycle.

Analysis of generalized processor-sharing systems with two classes of customers and exponential services

2004 ◽  
Vol 41 (3) ◽  
pp. 832-858 ◽  
Author(s):  
Fabrice Guillemin ◽  
Didier Pinchon
Keyword(s):  
Hilbert Problem ◽  
Probability Distributions ◽  
Probability Generating Function ◽  
Joint Probability ◽  
Processor Sharing ◽  
Generalized Processor Sharing ◽  
Quarter Plane ◽  
Explicit Formulae ◽  
Gps System ◽  
Number Of Customers

We derive in this paper closed formulae for the joint probability generating function of the number of customers in the two FIFO queues of a generalized processor-sharing (GPS) system with two classes of customers arriving according to Poisson processes and requiring exponential service times. In contrast to previous studies published on the GPS system, we show that it is possible to establish explicit expressions for the generating functions of the number of customers in each queue without calling for the formulation of a Riemann–Hilbert problem. We specifically prove that the problem of determining the unknown functions due to the reflecting conditions on the boundaries of the positive quarter plane can be reduced to a Poisson equation. The explicit formulae are then used to derive some characteristics of the GPS system (in particular the tails of the probability distributions of the numbers of customers in each queue).

scholarly journals A Perishable Inventory System with Postponed Demands and Multiple Server Vacations

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
R. Jayaraman ◽  
B. Sivakumar ◽  
G. Arivarignan
Keyword(s):  
Steady State ◽  
Joint Probability ◽  
Life Time ◽  
Inventory System ◽  
Inventory Level ◽  
Joint Probability Distribution ◽  
Single Server ◽  
Cost Rate ◽  
Stochastic Inventory ◽  
Number Of Customers

A mathematical modelling of a continuous review stochastic inventory system with a single server is carried out in this work. We assume that demand time points form a Poisson process. The life time of each item is assumed to have exponential distribution. We assume(s,S)ordering policy to replenish stock with random lead time. The server goes for a vacation of an exponentially distributed duration at the time of stock depletion and may take subsequent vacation depending on the stock position. The customer who arrives during the stock-out period or during the server vacation is offered a choice of joining a pool which is of finite capacity or leaving the system. The demands in the pool are selected one by one by the server only when the inventory level is aboves, with interval time between any two successive selections distributed as exponential with parameter depending on the number of customers in the pool. The joint probability distribution of the inventory level and the number of customers in the pool is obtained in the steady-state case. Various system performance measures in the steady state are derived, and the long-run total expected cost rate is calculated.

scholarly journals Steady-State Analysis of a Flexible Markovian Queue with Server Breakdowns

Entropy ◽  
2019 ◽  
Vol 21 (3) ◽  
pp. 259 ◽  
Author(s):  
Messaoud Bounkhel ◽  
Lotfi Tadj ◽  
Ramdane Hedjar
Keyword(s):  
Steady State ◽  
Probability Generating Function ◽  
Threshold Level ◽  
System Size ◽  
Single Server ◽  
Markovian Queue ◽  
Breakdown Rates ◽  
Bulk Service ◽  
Service Rates ◽  
Number Of Customers

A flexible single-server queueing system is considered in this paper. The server adapts to the system size by using a strategy where the service provided can be either single or bulk depending on some threshold level c. If the number of customers in the system is less than c, then the server provides service to one customer at a time. If the number of customers in the system is greater than or equal to c, then the server provides service to a group of c customers. The service times are exponential and the service rates of single and bulk service are different. While providing service to either a single or a group of customers, the server may break down and goes through a repair phase. The breakdowns follow a Poisson distribution and the breakdown rates during single and bulk service are different. Also, repair times are exponential and repair rates during single and bulk service are different. The probability generating function and linear operator approaches are used to derive the system size steady-state probabilities.

scholarly journals On the Nonsymmetric Longer Queue Model: Joint Distribution, Asymptotic Properties, and Heavy Traffic Limits

2013 ◽  
Vol 2013 ◽  
pp. 1-21 ◽  
Author(s):  
Charles Knessl ◽  
Haishen Yao
Keyword(s):  
Heavy Traffic ◽  
Asymptotic Properties ◽  
Joint Probability ◽  
Integral Representations ◽  
Joint Probability Distribution ◽  
Single Server ◽  
Poisson Arrival ◽  
Large Numbers ◽  
Node Network ◽  
Number Of Customers

We consider two parallel queues, each with independent Poisson arrival rates, that are tended by a single server. The exponential server devotes all of its capacity to the longer of the queues. If both queues are of equal length, the server devotesνof its capacity to the first queue and the remaining1−νto the second. We obtain exact integral representations for the joint probability distribution of the number of customers in this two-node network. Then we evaluate this distribution in various asymptotic limits, such as large numbers of customers in either/both of the queues, light traffic where arrivals are infrequent, and heavy traffic where the system is nearly unstable.

Sign in / Sign up

Export Citation Format

Share Document