Availability Formulas and Performance Measures for Separable Degradable Networks

Abstract

We derive explicit formulas for availability and performance measures in complex degradable stochastic networks. The networks consist of interacting service systems (queues) where servers are unreliable and may break down. The breakdown and repair mechanisms are of rather general structure, and may show complex dependencies among the nodes of the networks. The degradable network models investigated incorporate the breakdowns of the servers and their repairs as well as the behavior of the service mechanisms and the customers' routing into the overall Markovian system description. The service rates may depend on the local loads of the service centers, and furthermore working and repair periods are globally determined by state dependent intensities. Three different rules are applied to handle the customers' routing connected with nodes in repair status. These rules originate from communication and production theory where they are used to resolve blocking situations.Modeling unreliable networks in this way leads to steady state distributions of product form for the supplemented Markovian state processes. From these first quantities we are able to derive point availabilities for single nodes and for groups of nodes as well as the standard performance measures for service networks.

Analysis of Jackson networks with infinite supply and unreliable nodes

“…Construction of rerouting schemes R I follows [16], and we distinguish the following rerouting schemes: 1. Skipping If a customer selects a down node as destination, the customer jumps to this node, spends no time there and immediately performs the next jump according to routing regime R until he arrives at a node in up status or leaves the network.…”

“…Breakdowns of nodes in standard Jackson networks were investigated in [16] and [13]. It turns out that breakdown of nodes with infinite supply requires a more specific regime to control breakdown and repair.…”

“…We have to show that the distribution given by (16) Under the required condition of either (8) and (9) solves balance equations (23). The last step of proving the theorem is by normalizingπ , which is possible because η i < μ i holds for all i ∈ W .…”

“…In production processes, an additional inventory with raw material guarantees that the machine will not idle even when there is sometimes no external demand. (II) Breakdown and repair [16] of servers with (i) simultaneous breakdown of groups of servers and (ii) group repair strategies. For example, repairing several servers simultaneously may lead to more efficient repair actions and thus reduce repair time.…”

Discrete time queueing networks with product form steady state. Availability and performance analysis in an integrated model

“…The resulting model is a pure queueing network where the unreliability of the servers is represented by slowing down the service rates. Integrated models for jointly investigating performance and availability in continuous time models are to be found, for example, in [39][40][41].…”