scholarly journals Obtaining derivatives of stationary queue length probabilities through the matrix-geometric solution method

1992 ◽  
Vol 5 (1) ◽  
pp. 69-73 ◽  
Author(s):  
Wei-Bo Gong ◽  
Jie Pan ◽  
Christos G. Cassandras
Keyword(s):  
Queue Length ◽  
Solution Method ◽  
Stationary Queue Length ◽  
The Matrix ◽  
Matrix Geometric Solution ◽  
Derivatives Of

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 the GI/M/1 Queue with Vacations and Multiple Service Phases

2017 ◽  
Vol 2017 ◽  
pp. 1-14
Author(s):  
Jianjun Li ◽  
Liwei Liu
Keyword(s):  
Performance Measures ◽  
Waiting Time ◽  
Solution Method ◽  
System Size ◽  
Mean Waiting Time ◽  
The Matrix ◽  
Matrix Geometric Solution ◽  
Semi Markov Process ◽  
Stationary Waiting Time ◽  
Number Of Customers

This paper considers a GI/M/1 queue with vacations and multiple service phases. Whenever the system becomes empty, the server takes a vacation, causing the system to move to vacation phase 0. If the server returns from a vacation to find no customer waiting, another vacation begins. Otherwise, the system jumps from phase 0 to some service phase i with probability qi,  i=1,2,…,N. Using the matrix geometric solution method and semi-Markov process, we obtain the distributions of the stationary system size at both arrival and arbitrary epochs. The distribution of the stationary waiting time of an arbitrary customer is also derived. In addition, we present some performance measures such as mean waiting time of an arbitrary customer, mean length of the type-i cycle, and mean number of customers in the system at the end of phase 0. Finally, some numerical examples are presented.

Analysis of the GI/Geo/c Queue with Working Vacations

2012 ◽  
Vol 197 ◽  
pp. 534-541
Author(s):  
Yan Gao ◽  
Wen Fen Liu
Keyword(s):  
Markov Chain ◽  
Optical Networks ◽  
Queue Length ◽  
Solution Method ◽  
Working Vacation ◽  
Modeling And Analysis ◽  
Matrix Geometric Solution ◽  
Working Vacations ◽  
The Stability ◽  
Irreducible Markov Chain

Working vacation queue models are well applied in the modeling and analysis of the router in optical networks. The GI/Geo/c queue with working vacations is studied in this paper. Through establishing two-dimensional Markov chain and using matrix-geometric solution method, the stability condition is derived. Adopting UL-type RG-factorization of irreducible Markov chain, the stationary distribution is given. Based on these, the probability distribution of queue-length and PGF of waiting time are obtained in the end.

Modeling of Distributed Parameter Processes with Neural Networks

2014 ◽  
Vol 555 ◽  
pp. 530-540
Author(s):  
Honoriu Vălean ◽  
Mihail Abrudean ◽  
Mihaela Ligia Ungureşan ◽  
Iulia Clitan ◽  
Vlad Mureşan
Keyword(s):  
Neural Networks ◽  
Simulation Method ◽  
Distributed Parameter ◽  
Hearth Furnace ◽  
Original Solution ◽  
Steel Pipes ◽  
The Matrix ◽  
The Neural Networks ◽  
Derivatives Of ◽  
Seamless Steel

In this paper an original solution for the modeling of distributed parameter processes using neural networks is presented. The proposed method represents a particular alternative to a very accurate modeling-simulation method for this kind of processes, the method based on the matrix of partial derivatives of the state vector (Mpdx), associated with Taylor series. In order to compare the performances generated by the two methods, a distributed parameter thermal process associated to a rotary hearth furnace (R.H.F) from the technological flow of producing seamless steel pipes is considered. The main similarities and differences between the two methods are highlighted in the paper. The treated solution represents a premise for the usage of the neural networks in the automatic control of the distributed parameter processes domain.

scholarly journals Steady-state analysis of a multiserver queue in the Halfin-Whitt regime

2008 ◽  
Vol 40 (2) ◽  
pp. 548-577 ◽  
Author(s):  
David Gamarnik ◽  
Petar Momčilović
Keyword(s):  
Queue Length ◽  
Heavy Traffic ◽  
Length Distribution ◽  
Moment Generating Function ◽  
Compact Representation ◽  
Single Server ◽  
Service Time Distribution ◽  
Spare Capacity ◽  
Queue Length Distribution ◽  
Stationary Queue Length

We consider a multiserver queue in the Halfin-Whitt regime: as the number of serversngrows without a bound, the utilization approaches 1 from below at the rateAssuming that the service time distribution is lattice valued with a finite support, we characterize the limiting scaled stationary queue length distribution in terms of the stationary distribution of an explicitly constructed Markov chain. Furthermore, we obtain an explicit expression for the critical exponent for the moment generating function of a limiting stationary queue length. This exponent has a compact representation in terms of three parameters: the amount of spare capacity and the coefficients of variation of interarrival and service times. Interestingly, it matches an analogous exponent corresponding to a single-server queue in the conventional heavy-traffic regime.

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.

Symmetric queues with batch departures and their networks

1996 ◽  
Vol 28 (01) ◽  
pp. 308-326 ◽  
Author(s):  
Masakiyo Miyazawa ◽  
Ronald W. Wolff
Keyword(s):  
Queue Length ◽  
Production Systems ◽  
Stochastic Order ◽  
Length Distribution ◽  
Stability Conditions ◽  
Service Discipline ◽  
Single Node ◽  
Stationary Queue Length ◽  
Poisson Arrivals ◽  
Service Times

Batch departures arise in various applications of queues. In particular, such models have been studied recently in connection with production systems. For the most part, however, these models assume Poisson arrivals and exponential service times; little is known about them under more general settings. We consider how their stationary queue length distributions are affected by the distributions of interarrival times, service times and departing batch sizes of customers. Since this is not an easy problem even for single departure models, we first concentrate on single-node queues with a symmetric service discipline, which is known to have nice properties. We start with pre-emptive LIFO, a special case of the symmetric service discipline, and then consider symmetric queues with Poisson arrivals. Stability conditions and stationary queue length distributions are obtained for them, and several stochastic order relations are considered. For the symmetric queues and Poisson arrivals, we also discuss their network. Stability conditions and the stationary joint queue length distribution are obtained for this network.

An Applied Form of Kane’s Equations of Motions

2005 ◽  
Author(s):  
Hossein Nejat Pishkenari ◽  
Amir Lotfi Gaskarimahalle ◽  
Seyed Babak Ghaemi Oskouei ◽  
Ali Meghdari
Keyword(s):  
Degrees Of Freedom ◽  
Control Unit ◽  
Generalized Coordinates ◽  
Kane’S Equations ◽  
Kane's Equations ◽  
The Matrix ◽  
Angular Velocities ◽  
New Form ◽  
Derivatives Of ◽  
Equations Of Motions

In this paper we have presented a new form of Kane’s equations. This new form is expressed in the matrix form with the components of partial derivatives of linear and angular velocities relative to the generalized speeds and generalized coordinates. The number of obtained equations is equal to the number of degrees of freedom represented in a closed form. Also the equations can be rearranged to appear only one of the time derivatives of generalized speeds in each equation. This form is appropriate especially when one intends to derive equations recursively. Hence in addition to the simplicity, the amount of calculations is noticeably reduced and also can be used in a control unit.

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