Decay properties and quasi-stationary distributions for stopped Markovian bulk-arrival and bulk-service queues

Abstract

International audienceWe consider decay properties including the decay parameter, invariant measures and quasi-stationary distributions for a Markovian bulk-arrival and bulk-service queue which stops when the waiting line is empty. Investigating such a model is crucial for understanding the busy period and other related properties of the Markovian bulk-arrival and bulk-service queuing processes. The exact value of the decay parameter is first obtained. We show that the decay parameter can be easily expressed explicitly. The invariant measures and quasi-distributions are then revealed. We show that there exists a family of invariant measures indexed by ∈[0,]. We then show that under some mild conditions there exists a family of quasi-stationary distributions also indexed by ∈[0,]. The generating functions of these invariant measures and quasi-stationary distributions are presented. We further show that this stopped Markovian bulk-arrival and bulk-service queueing model is always -transient. Some deep properties regarding -transience are examined and revealed. The clear geometric interpretation of the decay parameter is explained. A few examples are then provided to illustrate the results obtained in this paper

Uniqueness and Decay Properties of Markov Branching Processes with Disasters

“…k=1 m k q kj = −μm j , j ≥ 1, or, by noting the structure of the BWD q-matrix Q, j +1 k=1 m k kb j −k+1 + m j (−ja) = −μm j , j ≥ 1, (3.11)where μ = (−1)G (q) denotes the decay parameter. By(3.11), it is easily seen that, similarly as proved in[5] (see the proof of Theorem 5.1 therein), the generating function of this unique λ C -invariant measure M(s) = ∞ j =1 m j s j is convergent at least in a sufficiently small interval (−τ, τ ). Multiplying s j on (3.11) and summing j from1 to ∞ for s ∈ (−τ, τ ) immediately yields ∞ j =1 j +1 k=1 m k kb j −k+1 s j − a ∞ j =1 jm j s j = −μ ∞ j =1 m j s j .Applying Fubini's theorem to the first term on the left-hand side of the above equality together with a trivial algebra yields∞ j =0 b j s j − as ∞ k=1 km k s k−1 + μ ∞ k=1 m k s k = m 1 b 0 , i.e.…”

“…or, by noting the structure of the BWD q-matrix Q, j +1 k=1 m k kb j −k+1 + m j (−ja) = −μm j , j ≥ 1, (3.11) where μ = (−1)G (q) denotes the decay parameter. By (3.11), it is easily seen that, similarly as proved in [5] (see the proof of Theorem 5.1 therein), the generating function of this unique λ C -invariant measure M(s) = ∞ j =1 m j s j is convergent at least in a sufficiently small interval (−τ, τ ). Multiplying s j on (3.11) and summing j from 1 to ∞ for s ∈ (−τ, τ ) immediately yields…”

Quasi-Stationary Distributions

“…For diffusions and other continuous-state processes, a good starting point is Steinsaltz and Evans [140] (but see also Cattiaux et al [22] and Pinsky [119]) and for branching processes there is an excellent recent review by Lambert [95,Section 3]. Whilst many issues remain unresolved, the theory has reached maturity, and the use of quasi-stationary distributions is now widespread, encompassing varied and contrasting areas of application, including cellular automata (Atman and Dickman [9]), complex systems (Collet et al [34]), ecology (Day and Possingham [41], Gosselin [63], Gyllenberg and Sylvestrov [68], Kukhtin et al [89], Pollett [122]) epidemics (Nåsell [106,107,108], Artalejo et al [6,7]), immunology (Stirk et al [141]), medical decision making (Chan et al [24]), physical chemistry (Dambrine and Moreau [37,38], Oppenheim et al [112], Pollett [121]), queues (Boucherie [17], Chen et al [25], Kijima and Makimoto [84]), reliability (Kalpakam and Shahul-Hameed [73], Kalpakam [74], Li and Cao [98,99]), survival analysis (Aalen and Gjessing [1,2], Steinsaltz and Evans [139]) and telecommunications (Evans [53], Ziedins [152]).…”

“…Moreover, since the rates of convergence of P i (X(t) = j) and P i (T > t) must be equal, we have α = α 0 . Finally, by the argument given in the proof of The existence of the limits in (25) has been proven in various settings, usually more restricted, however, than those required for the existence of a quasistationary distribution (see, for example, [131,30]).…”

Decay Properties of n-type Markov Branching Processes with Disasters

“…For more recent developments, one can refer to the survey paper of Van Doorn and Pollett [18]. For the related works, see [6,7,[15][16][17]21].…”