Heterogeneous Finite-Source Retrial Queues

Abstract

In this paper we investigate a single server retrial queue with a finite number of heterogeneous sources of calls. It is assumed when a given source is idle it will generate a primary call after an exponentially distributed time. If the server is free at the time of the request's arrival then the call starts to be served. The service time is also exponentially distributed. During the service time the source cannot generate a new primary call. After service the source moves into free state and can generate a new call again. If the server is busy at the time of the arrival of a primary call, then the source starts generating so called repeated calls with exponentially distributed times until it finds the server free. As before, after service the source becomes free and can generate a new primary call again. We assume that the primary calls, repeated attempts and service times are mutually independent. This queueing system and its variants could be used to model magnetic disk memory systems, local area networks with CSMA/CD protocols and collision avoidance local area networks. The novelty of this model is the heterogeneity of the calls, which means that each call is characterized by its own arrival, repeated and service rates. The aim of the paper is to give the usual steady-state performance measures of the system. To do so, an efficient software tool, MOSEL ( Modeling, Specification and Evaluation Language ) developed at the University of Erlangen, Germany, is used to formulate and solve the problem. Several sample numerical results illustrate the power of the tool showing the effect of different parameters on the system measures.

The Impact of Retrials on the Performance of Self-Organizing Systems

“…It can be seen that the parameter p does not only have an influence on the values of the mean response time but also on the shape of the graph. The maximum arising in the mean response time of finite-source retrial queues has been noticed by other researchers already (see, e.g., [17,18]). However, to the best of our knowledge, no thorough explanation has been given for it, yet.…”

Modeling Wireless Sensor Networks Using Finite-Source Retrial Queues with Unreliable Orbit

“…1 Due to harsh resource constraints and the resulting limited transmission range, each node i (located at (x i , y i )) is only able to communicate directly with its immediate neighbors, i.e., with all nodes j where |x j − x i | ≤ 1 and |y j − y i | ≤ 2. For example, the node located at (6, 4) is able to exchange messages directly with the nodes located at (6, 6), (5,5), (5,3), (6,2), (7,3), and also with the sink. We further assume that each node is aware of its own distance to the sink measured in the number of hops.…”

Congestion Probabilities in Erlang-Engset Multirate Loss Models under the Multiple Fractional Channel Reservation Policy

“…In addition, the R-MFCR model is extended by assuming that calls are generated by finite sources. This is the well-known quasi-random process (e.g., [1,2,6,8,14,24,33,48]). The springboard for the analysis of quasirandom traffic in a multirate loss system can be considered the Engset Multirate Loss Model (EnMLM) proposed in [33].…”

“…where: j * = K k=1 c k y * k (j) and l k (j −c k ) is given by (2). The CBPs can now be obtained via (3).…”