Markov Renewal Methods in Restart Problems in Complex Systems

Abstract

A task with ideal execution time L such as the execution of a computer program or the transmission of a file on a data link may fail, and the task then needs to be restarted. The task is handled by a complex system with features similar to the ones in classical reliability: failures may be mitigated by using server redundancy in parallel or k-out-of-n arrangements, standbys may be cold or warm, one or more repairmen may take care of failed components, etc. The total task time X (including restarts and pauses in failed states) is investigated with particular emphasis on the tail P(X > x). A general alternating Markov renewal model is proposed and an asymptotic exponential form P(X > x) ∼ Ce −γx identified for the case of a deterministic task time L ≡ . The rate γ is given by equating the spectral radius of a certain matrix to 1, and the asymptotic form of γ = γ( ) as → ∞ is derived, leading to the asymptotics of P(X > x) for random task times L. A main finding is that X is always heavy-tailed if L has unbounded support. The case where the Markov renewal model is derived by lumping in a continuous-time finite Markov process with exponential holding times is given special attention, and the study includes analysis of the effect of processing rates that differ with state or time.

Time inhomogeneity in longest gap and longest run problems

“…Remark 6.1. Motivated by problems such as transmitting files of randomly fluctuating sizes, a main problem in [3,5] is to replace by a random L independently of N . It is assumed that L is drawn once and fixed forever which corresponds to the RESTART problem.…”

“…Section 6 deals with what provided our initial motivation, the study of the tail of the total execution time X of a task like program or file transmission in a fault-tolerant computing environment working under the restart protocol, where the task needs to be completely restarted after failure. Here takes the role of the ideal task time, failures occur at the epochs of N and so X = + D. Earlier studies of similar problems are in Asmussen et al [3,4,5] and Jelenković et al [13,12]. The novelty here is the time-inhomogeneity.…”

“…Time-inhomogeneity only seems to have been studied in the framework of Markovian regime switching which is somewhat different from the models of this paper by being asymptotically stationary rather than exhibiting a trend. Some references are [1,11,6,5] ( [6] also contains some early and in part unprecise version of a few of the results of this paper).…”

Markov dependence in renewal equations and random sums with heavy tails

“…. In branching processes, it occurs when individuals have several types, and in insurance and finance, it relates to regime-switching; another relevant example comes from computer reliability problems, [11]. The known asymptotic results on the Z i (x) depend crucially on the spectral radius ρ of P : if ρ = 1, in particular if P is stochastic, limits exists whereas otherwise the order is exponential, e γx , where γ > 0 when ρ > 1 and γ < 0 when ρ < 1 and the F ij are light-tailed [again, regularity conditions are required].…”

Modelling failures times with dependent renewal type models via exchangeability

“…Applications include: occurrence of rare events [4], streams of customers [5], lifetimes modeling [6,7], disease activity [8] and web applications [9], among others. Theoretical contributions in some specific dependent renewal-type models, including Markov renewal processes, can be found in [10,11,12,13]. For a classical account of the theory of Markov renewal processes, we refer the reader to [14].While some of these models provide with excellent extensions, most apply only for specific failure distributions.…”