Large Closed Queueing Networks in Semi-Markov Environment and Their Application (original) (raw)
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A Large Closed Queueing Network in Markov Environment and Its Application
2006
A paper studies a closed queueing network containing a server station and k client stations. The server station is an infinite server queueing system, and client stations are single server queueing systems with autonomous service, i.e. every client station serves customers (units) only at random instants generated by strictly stationary and ergodic sequence of random variables. The total number of units in the network is N . The expected times between departures in client stations are (Nμj ) . After service completion in the server station a unit is transmitted to the jth client station with probability pj (j = 1, 2, . . . , k), and being processed in the jth client station the unit returns to server station. The network is assumed to be in Markov environment. The Markov environment is defined by initial state, and phase space of dimension d. Then the routing matrix pj as well as transmission rates (which are expressed via parameters of the network) depend on the Markov state of the...
Queueing Systems, 2004
The paper studies a closed queueing network containing two types of node. The first type (server station) is an infinite server queueing system, and the second type (client station) is a single server queueing system with autonomous service, i.e. every client station serves customers (units) only at random instants generated by strictly stationary and ergodic sequence of random variables. It is assumed that there are r server stations. At the initial time moment all units are distributed in the server stations, and the ith server station contains N i units, i = 1, 2,. .. , r, where all the values N i are large numbers of the same order. The total number of client stations is equal to k. The expected times between departures in the client stations are small values of the order O(N −1) (N = N 1 + N 2 + • • • + N r). After service completion in the ith server station a unit is transmitted to the j th client station with probability p i,j (j = 1, 2,. .. , k), and being served in the j th client station the unit returns to the ith server station. Under the assumption that only one of the client stations is a bottleneck node, i.e. the expected number of arrivals per time unit to the node is greater than the expected number of departures from that node, the paper derives the representation for non-stationary queue-length distributions in non-bottleneck client stations.
An infinite-server queue influenced by a semi-Markovian environment
Queueing Systems, 2008
We consider an infinite-server queue, where the arrival and service rates are both governed by a semi-Markov process that is independent of all other aspects of the queue. In particular, we derive a system of equations that are satisfied by various "parts" of the generating function of the steady-state queue-length, while assuming that all arrivals bring an amount of work to the system that is either Erlang or hyperexponentially distributed. These equations are then used to show how to derive all moments of the steady-state queue-length. We then conclude by showing how these results can be slightly extended, and used, along with a transient version of Little's law, to generate rigorous approximations of the steady-state queue-length in the case that the amount of work brought by a given arrival is of an arbitrary distribution.
2009
In this paper the MM K k=1 CP P k /GE/c/L G-queue is introduced and proposed as a generalised Markovian node model in telecommunications networks. An exact and computationally efficient solution is obtained for the steady-state probabilities and performance measures. Issues concerning the computational effort are also discussed. The proposed queue is applied to the performance analysis of optical burst switching (OBS) nodes. The numerical results obtained and also the numerical results of a previous model are compared to the simulation results of the OBS obtained using captured traffic traces. We have also introduced negative customers into the model, in an innovative way, in order to account for the loss of packets due to technology limitations of the FDL's (fiber delay loop), which is rather specific to the optical domain. The model is quite promising as a viable performance predictor.
On queues with Markov modulated service rates
Queueing Systems, 2005
In this paper, we consider a queue whose service speed changes according to an external environment that is governed by a Markov process. It is possible that the server changes its service speed many times while serving a customer. We derive first and second moments of the service time of customers in system using first step analysis to obtain an insight on the service process. In fact, we obtain an intriguing result in that the moments of service time actually depend on the arrival process! We also show that the mean service rate is not the reciprocal of the mean service time. Further, since it is not possible to obtain a closed form expression for the queue length distribution, we use matrix geometric methods to compute performance measures such as average queue length and waiting time. We apply the method of large deviations to obtain tail distributions of the workload in the queue using the concept of effective bandwidth. We present two applications in computer systems: 1) Web server with multi-class requests and 2) CPU with multiple processes. We illustrate the analysis and various methods discussed with the help of numerical examples for the above two applications.
An analytical study of various telecomminication networks using markov models
IOP Conference Series: Materials Science and Engineering, 2015
The main aim of this paper is to examine issues relating to the performance of various Telecommunication networks, and applied queuing theory for better design and improved efficiency. Firstly, giving an analytical study of queues deals with quantifying the phenomenon of waiting lines using representative measures of performances, such as average queue length (on average number of customers in the queue), average waiting time in queue (on average time to wait) and average facility utilization (proportion of time the service facility is in use). In the second, using Matlab simulator, summarizes the finding of the investigations, from which and where we obtain results and describing methodology for a) compare the waiting time and average number of messages in the queue in M/M/1 and M/M/2 queues b) Compare the performance of M/M/1 and M/D/1 queues and study the effect of increasing the number of servers on the blocking probability M/M/k/k queue model. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
Computers & Industrial Engineering, 2006
The independence of processes in queueing systems is generally assumed when developing queueing models. However, real systems often involve several process dependencies, and failure to take these into consideration can lead to serious underestimation of the performance measures. We consider herein a single server queueing system with a Markov renewal process (MRP) for its arrival process and a general service time distribution, and derive the distribution function and correlation coefficient of the departure process. Since the departure process also often corresponds to an arrival process in downstream queues, the results obtained here can be used to derive a better approximation of the performance measures of a non-product form general queueing network.