Loss systems in a random environment: steady state analysis

Abstract

We consider a single server system with infinite waiting room in a random environment. The service system and the environment interact in both directions. Whenever the environment enters a prespecified subset of its state space the service process is completely blocked: Service is interrupted and newly arriving customers are lost. We prove a product-form steady state distribution of the joint queueingenvironment process. A consequence is a strong insensitivity property for such systems. We discuss several applications, for example, from inventory theory and reliability theory, and show that our result extends and generalizes several theorems found in the literature, for example, of queueing-inventory processes. We investigate further classical loss systems, where, due to finite waiting room, loss of customers occurs. In connection with loss of customers due to blocking by the environment and service interruptions new phenomena arise.Keywords Queueing systems · Random environment · Product form steady state · Loss systems · M/M/1/∞ · M/M/m/∞ · M/M/m/N · Inventory systems · Availability · Lost sales Mathematics Subject Classification Primary 60K · 60J10 · 60F05 · 60K20 · 90B22 · 90B05

Analysis of a Multiserver Queueing-Inventory System

“…Then the components of the steady-state probability vector of the process {X( ) | ≥ 0} with generator matrix W 1 are ( ) = −1 Θ , ≥ 0, 0 ≤ ≤ , the probabilities correspond to the distribution of number of customers in the system as given in (12), and the probabilities are obtained in (15).…”

“…Krishnamoorthy and Viswanath [6] subsume Schwarz et al [5] by extending the latter to production inventory with positive service time. References [7] of Sivakumar and Arivarignan, [8] of Krishnamoorthy and Narayanan, [9] of Deepak et al, [10] of Schwarz and Daduna, [11] of Schwarz et al, and [12] of Krenzler and Daduna are a few other significant contributions to inventory with positive service time. Protection of production and service stages in a queueing-inventory model, with Erlang distributed service and interproduction time, is analyzed by Krishnamoorthy et al [13].…”

“…The authors derive joint stationary distribution of queue length and on-hand inventory at various stations in explicit product form. A recent contribution of interest to inventory with positive service time involving a random environment is by Krenzler and Daduna [15] wherein also a stochastic decomposition of the system is established. They prove a necessary and sufficient condition for a product form steady-state distribution of the joint queueing-environment process to exist.…”

“…They prove a necessary and sufficient condition for a product form steady-state distribution of the joint queueing-environment process to exist. A still more recent paper by Krenzler and Daduna [12] investigates inventory with positive service time in a random environment embedded in a Markov chain. They provide a counter example to show that the steadystate distribution of an / /1/∞ system with ( , ) policy and lost sales need not have a product form.…”

Modeling and Performance Analysis of a Node in Fault Tolerant Wireless Sensor Networks

“…A survey on related queueing-inventory systems is [8]. Recent results on single nodes are in [6], [7], with more relevant references.…”