Analysis of State Dependent Vacation Queues with Threshold Gated Service Policy

2012 ◽  
pp. 274-291
Author(s):  
Yew Sing
Keyword(s):  
Performance Measures ◽  
Cycle Time ◽  
Busy Period ◽  
Simple Approach ◽  
Second Moment ◽  
State Dependent ◽  
The Mean ◽  
Service Policy ◽  
Set Up ◽  
Number Of Customers

In this article, the authors introduce a simple approach for modeling and analyzing a queue where the server may take repeated vacations. When a busy period ends, the server takes a vacation of random duration. At the end of each vacation, the server may either start a new vacation or resume service. If a queue is found of less than customers, the server will always take a new vacation. If there are at least customers in queue, the server provides services to those customers after a brief set-up time. The authors obtain several performance measures of the system, including the mean and second moment of the cycle time, the number of customers in a cycle of service, and the expected delay experienced by a customer.

Analysis of State Dependent Vacation Queues with Threshold Gated Service Policy

2010 ◽  
Vol 1 (2) ◽  
pp. 32-49 ◽  
Author(s):  
Yew Sing
Keyword(s):  
Performance Measures ◽  
Cycle Time ◽  
Busy Period ◽  
Simple Approach ◽  
Second Moment ◽  
State Dependent ◽  
The Mean ◽  
Service Policy ◽  
Set Up ◽  
Number Of Customers

In this article, the authors introduce a simple approach for modeling and analyzing a queue where the server may take repeated vacations. When a busy period ends, the server takes a vacation of random duration. At the end of each vacation, the server may either start a new vacation or resume service. If a queue is found of less than customers, the server will always take a new vacation. If there are at least customers in queue, the server provides services to those customers after a brief set-up time. The authors obtain several performance measures of the system, including the mean and second moment of the cycle time, the number of customers in a cycle of service, and the expected delay experienced by a customer.

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.

scholarly journals Hitting times of Markov chains, with application to state-dependent queues

1977 ◽  
Vol 17 (1) ◽  
pp. 97-107 ◽  
Author(s):  
R.L. Tweedie
Keyword(s):  
Markov Chain ◽  
Waiting Time ◽  
Busy Period ◽  
Hitting Times ◽  
Arrival Process ◽  
General State ◽  
State Dependent ◽  
General State Space ◽  
The Mean ◽  
Number Of Customers

We present in this note a useful extension of the criteria given in a recent paper [Advances in Appl. Probability 8 (1976), 737–771] for the finiteness of hitting times and mean hitting times of a Markov chain on sets in its (general) state space. We illustrate our results by giving conditions for the finiteness of the mean number of customers in the busy period of a queue in which both the service-times and the arrival process may depend on the waiting time in the queue. Such conditions also suffice for the embedded waiting time chain to have a unique stationary distribution.

scholarly journals Bounds on the rate of convergence for one class of inhomogeneous Markovian queueing models with possible batch arrivals and services

2018 ◽  
Vol 28 (1) ◽  
pp. 141-154 ◽  
Author(s):  
Alexander Zeifman ◽  
Rostislav Razumchik ◽  
Yacov Satin ◽  
Ksenia Kiseleva ◽  
Anna Korotysheva ◽  
...  
Keyword(s):  
Rate Of Convergence ◽  
Queueing Systems ◽  
Approximation Error ◽  
Linear Operators ◽  
Upper And Lower Bounds ◽  
Logarithmic Norm ◽  
Batch Arrivals ◽  
State Dependent ◽  
The Mean ◽  
Number Of Customers

AbstractIn this paper we present a method for the computation of convergence bounds for four classes of multiserver queueing systems, described by inhomogeneous Markov chains. Specifically, we consider an inhomogeneous M/M/S queueing system with possible state-dependent arrival and service intensities, and additionally possible batch arrivals and batch service. A unified approach based on a logarithmic norm of linear operators for obtaining sharp upper and lower bounds on the rate of convergence and corresponding sharp perturbation bounds is described. As a side effect, we show, by virtue of numerical examples, that the approach based on a logarithmic norm can also be used to approximate limiting characteristics (the idle probability and the mean number of customers in the system) of the systems considered with a given approximation error.

On the busy-period distributions of M/G/1/K queues by state-dependent arrivals and FCFS/LCFS-P service disciplines

1985 ◽  
Vol 22 (04) ◽  
pp. 912-919
Author(s):  
J. George Shanthikumar ◽  
Ushio Sumita
Keyword(s):  
Busy Period ◽  
Stochastic Ordering ◽  
State Dependent ◽  
Service Disciplines ◽  
Recursion Formulas ◽  
Deterministic Service ◽  
Service Times ◽  
Number Of Customers ◽  
Stieltjes Transforms

The busy-period distributions of M/G/1/K queues with state-dependent arrival rates are considered. Two recursion formulas for the Laplace–Stieltjes transforms of the busy periods under the FCFS and preempt resume LCFS service disciplines are obtained. It is shown that the busy-period distributions for the two service disciplines are, in general, different, in contrast to the fact that they coincide for ordinary M/G/1 queues. For deterministic service times and arrival rates non-increasing in the number of customers in the system, stochastic ordering between these two busy periods is also established.

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.

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.

scholarly journals Linking Scheduling Criteria to Shop Floor Performance in Permutation Flowshops

Algorithms ◽  
2019 ◽  
Vol 12 (12) ◽  
pp. 263 ◽  
Author(s):  
Jose M. Framinan ◽  
Rainer Leisten
Keyword(s):  
Performance Measures ◽  
Cycle Time ◽  
Completion Time ◽  
Shop Floor ◽  
Due Dates ◽  
Processing Times ◽  
Work In Process ◽  
Set Up ◽  
The Impact ◽  
Scheduling Models

The goal of manufacturing scheduling is to allocate a set of jobs to the machines in the shop so these jobs are processed according to a given criterion (or set of criteria). Such criteria are based on properties of the jobs to be scheduled (e.g., their completion times, due dates); so it is not clear how these (short-term) criteria impact on (long-term) shop floor performance measures. In this paper, we analyse the connection between the usual scheduling criteria employed as objectives in flowshop scheduling (e.g., makespan or idle time), and customary shop floor performance measures (e.g., work-in-process and throughput). Two of these linkages can be theoretically predicted (i.e., makespan and throughput as well as completion time and average cycle time), and the other such relationships should be discovered on a numerical/empirical basis. In order to do so, we set up an experimental analysis consisting in finding optimal (or good) schedules under several scheduling criteria, and then computing how these schedules perform in terms of the different shop floor performance measures for several instance sizes and for different structures of processing times. Results indicate that makespan only performs well with respect to throughput, and that one formulation of idle times obtains nearly as good results as makespan, while outperforming it in terms of average cycle time and work in process. Similarly, minimisation of completion time seems to be quite balanced in terms of shop floor performance, although it does not aim exactly at work-in-process minimisation, as some literature suggests. Finally, the experiments show that some of the existing scheduling criteria are poorly related to the shop floor performance measures under consideration. These results may help to better understand the impact of scheduling on flowshop performance, so scheduling research may be more geared towards shop floor performance, which is sometimes suggested as a cause for the lack of applicability of some scheduling models in manufacturing.

Stability of Polling Networks with State-Dependent Server Routing

1995 ◽  
Vol 9 (4) ◽  
pp. 539-550 ◽  
Author(s):  
R. Schaβberger
Keyword(s):  
Markov Chain ◽  
Random Number ◽  
Necessary Condition ◽  
Finite Collection ◽  
Single Server ◽  
Positive Recurrence ◽  
State Dependent ◽  
Special Cases ◽  
Service Policy ◽  
Number Of Customers

The polling network considered here consists of a finite collection of stations visited successively by a single server according to a state-dependent routing scheme. At every visit of a station, a possibly random number of customers are served and depart from the network thereafter. The server takes a possibly random time to walk from one station to the next. The network receives groups of customers at Poisson instants. These customers are distributed randomly over the stations and wait until served. A necessary condition for stability is derived, which proves to be also sufficient in certain special cases. Such cases include a starlike network and a ringlike one with a greedy service policy. Mathematically, positive recurrence of a Markov chain is studied.

Characterizing losses during busy periods in finite buffer systems

2003 ◽  
Vol 40 (1) ◽  
pp. 242-249 ◽  
Author(s):  
Erol A. Peköz ◽  
Rhonda Righter ◽  
Cathy H. Xia
Keyword(s):  
Busy Period ◽  
Arrival Rate ◽  
Batch Size ◽  
Buffer Size ◽  
Service Rate ◽  
Finite Buffer ◽  
Initial Number ◽  
Poisson Arrivals ◽  
The Mean ◽  
Number Of Customers

For multiple-server finite-buffer systems with batch Poisson arrivals, we explore how the distribution of the number of losses during a busy period changes with the buffer size and the initial number of customers. We show that when the arrival rate equals the maximal service rate (ρ= 1), as the buffer size increases the number of losses in a busy period increases in the convex sense, and whenρ> 1, as the buffer size increases the number of busy period losses increases in the increasing convex sense. Also, the number of busy period losses is stochastically increasing in the initial number of customers. A consequence of our results is that, whenρ= 1, the mean number of busy period losses equals the mean batch size of arrivals regardless of the buffer size. We show that this invariance does not extend to general arrival processes.

Sign in / Sign up

Export Citation Format

Share Document