On conditional passage time structure of birth-death processes

1984 ◽  
Vol 21 (1) ◽  
pp. 10-21 ◽  
Author(s):  
Ushio Sumita
Keyword(s):  
Queueing System ◽  
First Passage Time ◽  
Arrival Rate ◽  
Passage Time ◽  
Death Process ◽  
Service Rate ◽  
Transition Rates ◽  
First Passage ◽  
Limiting Behavior ◽  
Birth Death

Let N(t) be a birth-death process on N = {0,1,2,· ··} governed by the transition rates λn > 0 (n ≧ 0) and μ η > 0 (n ≧ 1). Let mTm be the conditional first-passage time from r to n, given no visit to m where m <r < n. The downward conditional first-passage time nTm is defined similarly. It will be shown that , for any λn > 0 and μ η > 0. The limiting behavior of is considerably different from that of the ordinary first-passage time where, under certain conditions, exponentiality sets in as n →∞. We will prove that, when λn → λ > 0 and μ η → μ > 0 as n → ∞with ρ = λ /μ < 1, one has as r → ∞where TBP(λ,μ) is the server busy period of an M/M/1 queueing system with arrival rate λand service rate μ.

On conditional passage time structure of birth-death processes

1984 ◽  
Vol 21 (01) ◽  
pp. 10-21 ◽  
Author(s):  
Ushio Sumita
Keyword(s):  
Queueing System ◽  
First Passage Time ◽  
Arrival Rate ◽  
Passage Time ◽  
Death Process ◽  
Service Rate ◽  
Transition Rates ◽  
First Passage ◽  
Limiting Behavior ◽  
Birth Death

Let N(t) be a birth-death process on N = {0,1,2,· ··} governed by the transition rates λ n &gt; 0 (n ≧ 0) and μ η &gt; 0 (n ≧ 1). Let mTm be the conditional first-passage time from r to n, given no visit to m where m &lt;r &lt; n. The downward conditional first-passage time nTm is defined similarly. It will be shown that , for any λ n &gt; 0 and μ η &gt; 0. The limiting behavior of is considerably different from that of the ordinary first-passage time where, under certain conditions, exponentiality sets in as n →∞. We will prove that, when λ n → λ &gt; 0 and μ η → μ &gt; 0 as n → ∞with ρ = λ /μ &lt; 1, one has as r → ∞where T BP(λ,μ) is the server busy period of an M/M/1 queueing system with arrival rate λand service rate μ.

On limiting behavior of ordinary and conditional first-passage times for a class of birth-death processes

1987 ◽  
Vol 24 (1) ◽  
pp. 235-240 ◽  
Author(s):  
U. Sumita
Keyword(s):  
First Passage Time ◽  
Arrival Rate ◽  
Passage Time ◽  
First Passage Times ◽  
Death Process ◽  
Service Rate ◽  
First Passage ◽  
Limiting Behavior ◽  
Passage Times ◽  
Birth Death

Let N(t) be a birth-death process on 𝒩= {0, 1, 2, ·· ·} governed by the transition rates λn > 0 (n ≧ 0) and μn > 0 (n ≧ 1) where λn → λ > 0 and μn → μ > 0 as n → ∞ and ρ = λ/μ. Let Tmn be the first-passage time of N(t) from m to n and define It is shown that, when converges in distribution to TBP(μ,λ) as n → ∞ where TΒΡ (μ,λ) is the server busy period of an M/M/1 queueing system with arrival rate μ and service rate λ. Correspondingly T0n/E[T0n] converges to 1 with probability 1 as n →∞. Of related interest is the conditional first-passage time mTrn of N(t) from r to n given no visit to m where m < r < n. As we shall see, the conditional first-passage time of N(t) can be viewed as an ordinary first-passage time of a modified birth-death process M(t) governed by where are generated from λn and μn. Furthermore it is shown that for and while for and This enables one to establish the relation between the limiting behavior of the ordinary first-passage times and that of the conditional first-passage times.

On limiting behavior of ordinary and conditional first-passage times for a class of birth-death processes

1987 ◽  
Vol 24 (01) ◽  
pp. 235-240 ◽  
Author(s):  
U. Sumita
Keyword(s):  
First Passage Time ◽  
Arrival Rate ◽  
Passage Time ◽  
First Passage Times ◽  
Death Process ◽  
Service Rate ◽  
First Passage ◽  
Limiting Behavior ◽  
Passage Times ◽  
Birth Death

LetN(t) be a birth-death process on 𝒩= {0, 1, 2, ·· ·} governed by the transition ratesλn&gt; 0 (n≧ 0) andμn&gt; 0 (n≧ 1) whereλn→λ&gt; 0 andμn→μ&gt; 0 asn→ ∞ andρ=λ/μ. LetTmnbe the first-passage time ofN(t) frommtonand defineIt is shown that, whenconverges in distribution toTBP(μ,λ)asn → ∞whereTΒΡ (μ,λ)is the server busy period of anM/M/1 queueing system with arrival rateμand service rateλ. CorrespondinglyT0n/E[T0n] converges to 1 with probability 1 asn→∞. Of related interest is the conditional first-passage timemTrnofN(t) from r tongiven no visit tomwherem &lt; r &lt; n.As we shall see, the conditional first-passage time ofN(t) can be viewed as an ordinary first-passage time of a modified birth-death processM(t) governed bywhereare generated fromλnandμn. Furthermore it is shown that forandwhile forandThis enables one to establish the relation between the limiting behavior of the ordinary first-passage times and that of the conditional first-passage times.

Spectral structure of the first-passage-time densities for classes of Markov chains

1987 ◽  
Vol 24 (03) ◽  
pp. 631-643 ◽  
Author(s):  
Masaaki Kijima
Keyword(s):  
Markov Chains ◽  
First Passage Time ◽  
Passage Time ◽  
Death Process ◽  
Spectral Structure ◽  
First Passage ◽  
Completely Monotone ◽  
Spectral Representations ◽  
Absorbing States ◽  
Birth Death

Keilson [7] showed that for a birth-death process defined on non-negative integers with reflecting barrier at 0 the first-passage-time density from 0 to N (N to N + 1) has Pólya frequency of order infinity (is completely monotone). Brown and Chaganty [3] and Assaf et al. [1] studied the first-passage-time distribution for classes of discrete-time Markov chains and then produced the essentially same results as these through a uniformization. This paper addresses itself to an extension of Keilson's results to classes of Markov chains such as time-reversible Markov chains, skip-free Markov chains and birth-death processes with absorbing states. The extensions are due to the spectral representations of the infinitesimal generators governing these Markov chains. Explicit densities for those first-passage times are also given.

Spectral structure of the first-passage-time densities for classes of Markov chains

1987 ◽  
Vol 24 (3) ◽  
pp. 631-643 ◽  
Author(s):  
Masaaki Kijima
Keyword(s):  
Markov Chains ◽  
First Passage Time ◽  
Passage Time ◽  
Death Process ◽  
Spectral Structure ◽  
First Passage ◽  
Completely Monotone ◽  
Spectral Representations ◽  
Absorbing States ◽  
Birth Death

Keilson [7] showed that for a birth-death process defined on non-negative integers with reflecting barrier at 0 the first-passage-time density from 0 to N (N to N + 1) has Pólya frequency of order infinity (is completely monotone). Brown and Chaganty [3] and Assaf et al. [1] studied the first-passage-time distribution for classes of discrete-time Markov chains and then produced the essentially same results as these through a uniformization. This paper addresses itself to an extension of Keilson's results to classes of Markov chains such as time-reversible Markov chains, skip-free Markov chains and birth-death processes with absorbing states. The extensions are due to the spectral representations of the infinitesimal generators governing these Markov chains. Explicit densities for those first-passage times are also given.

Joint distributions of random variables and their integrals for certain birth-death and diffusion processes

1971 ◽  
Vol 3 (02) ◽  
pp. 339-352 ◽  
Author(s):  
J. Gani ◽  
D. R. Mcneil
Keyword(s):  
Diffusion Processes ◽  
First Passage Time ◽  
Integral Functional ◽  
Passage Time ◽  
Death Process ◽  
Traffic Jam ◽  
Joint Distributions ◽  
The Cost ◽  
And Storage ◽  
Birth Death

For the linear growth birth-death process with parameters λ n = nλ, μ n = nμ, Puri ((1966), (1968)) has investigated the joint distribution of the number X(t) of survivors in the process and the associated integral Y(t) = ∫0 t X(τ)dτ. In particular, he has obtained limiting results as t → ∞. Recently one of us (McNeil (1970)) has derived the distribution of the integral functional W x = ∫0 Tx g{X(τ)}dτ, where T x is the first passage time to the origin in a general birth-death process with X(0) = x and g(·) is an arbitrary function. Functionals of the form W x arise naturally in traffic and storage theory; for example W x may represent the total cost of a traffic jam, or the cost of storing a commodity until expiration of the stock. Moments of such functionals were found in the case of M/G/1 and GI/M/1 queues by Gaver (1969) and Daley (1969).

First-passage-time densities and avoiding probabilities for birth-and-death processes with symmetric sample paths

1998 ◽  
Vol 35 (2) ◽  
pp. 383-394 ◽  
Author(s):  
Antonio Di Crescenzo
Keyword(s):  
Queueing System ◽  
Transition Probabilities ◽  
First Passage Time ◽  
Sufficient Conditions ◽  
Passage Time ◽  
First Passage ◽  
Birth And Death Processes ◽  
Sample Paths ◽  
Reflecting Boundaries ◽  
Necessary And Sufficient

For truncated birth-and-death processes with two absorbing or two reflecting boundaries, necessary and sufficient conditions on the transition rates are given such that the transition probabilities satisfy a suitable spatial symmetry relation. This allows one to obtain simple expressions for first-passage-time densities and for certain avoiding transition probabilities. An application to an M/M/1 queueing system with two finite sequential queueing rooms of equal sizes is finally provided.

Joint distributions of random variables and their integrals for certain birth-death and diffusion processes

1971 ◽  
Vol 3 (2) ◽  
pp. 339-352 ◽  
Author(s):  
J. Gani ◽  
D. R. Mcneil
Keyword(s):  
Diffusion Processes ◽  
First Passage Time ◽  
Integral Functional ◽  
Passage Time ◽  
Death Process ◽  
Traffic Jam ◽  
Joint Distributions ◽  
The Cost ◽  
And Storage ◽  
Birth Death

For the linear growth birth-death process with parameters λn = nλ, μn = nμ, Puri ((1966), (1968)) has investigated the joint distribution of the number X(t) of survivors in the process and the associated integral Y(t) = ∫0tX(τ)dτ. In particular, he has obtained limiting results as t → ∞. Recently one of us (McNeil (1970)) has derived the distribution of the integral functional Wx = ∫0Txg{X(τ)}dτ, where Tx is the first passage time to the origin in a general birth-death process with X(0) = x and g(·) is an arbitrary function. Functionals of the form Wx arise naturally in traffic and storage theory; for example Wx may represent the total cost of a traffic jam, or the cost of storing a commodity until expiration of the stock. Moments of such functionals were found in the case of M/G/1 and GI/M/1 queues by Gaver (1969) and Daley (1969).

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