By E. A. van Doorn (auth.)

A stochastic approach {X(t): zero S t < =} with discrete nation house S c ~ is expounded to be stochastically expanding (decreasing) on an period T if the chances Pr{X(t) > i}, i E S, are expanding (decreasing) with t on T. Stochastic monotonicity is a simple structural estate for strategy behaviour. It provides upward push to significant bounds for numerous amounts comparable to the moments of the method, and gives the mathematical foundation for approximation algorithms. evidently, stochastic monotonicity turns into a extra tractable topic for research if the tactics into consideration are such that stochastic mono tonicity on an inter val zero < t < E implies stochastic monotonicity at the whole time axis. DALEY (1968) used to be the 1st to debate an identical estate within the context of discrete time Markov chains. regrettably, he known as this estate "stochastic monotonicity", it's extra acceptable, even though, to talk of strategies with monotone transition operators. KEILSON and KESTER (1977) have established the superiority of this phenomenon in discrete and non-stop time Markov strategies. They (and others) have additionally given an important and adequate situation for a (temporally homogeneous) Markov method to have monotone transition operators. even if such procedures could be stochas tically monotone as outlined above, now is dependent upon the preliminary country distribution. stipulations in this distribution for stochastic mono tonicity at the complete time axis to succeed got too via KEILSON and KESTER (1977).

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**Extra resources for Stochastic Monotonicity and Queueing Applications of Birth-Death Processes**

**Example text**

T , were h * < I and qk* tnttta veator S* = (* q_I' qo' q_1 for k > o n. This proaess is not stoahastiaally deareasing on any interval. 2. 2. •••• )T where qk - 0 for k > n. , En n = ~, then it is striatly stoahastiaally inareasing in the long run. , Enn < ~, then it is not stoahastiaally inareasing on any interval if E qiQi(x 2) < 0 • PROOF. 2) ~(t) T .. 1 this occurs for every i provided t is sufficiently large. Since qi .. 3. 1 yields for t sufficiently large o. 1 (i) and the fact that n are readily seen to imply that for fixed t > 0 and j sufficiently large e.

2 the problem is then whether the process is strictly stochastically inareasing on some interval or not. 38 5. • )T. n for some fixed i and all n. The process is denoted by {Xi(t)} .. 7). which. according to (4. I. is the ith row of the matrix E(t) .. (eij(t». e •• To determine whether {Xi(t)} is strictly stochastically increasing on an interval we argue as follows. Considering that qn = 0in' the sequence (qn"r n ) n is bounded. (O). -1. -1. where P * (t) is the transition matrix of the dual process (as usual an asterisk refers to the dual process).

With ~O=O, be the set of parameters of a natural birth-death process {X(t): 0,; t < .. } where 0 is a reflecting barrier. 9. e. , i = 0, I, ••.. 4 will be used. 9. 9..!. 15) and Q and.!. are the column vectors consisting of O's and l's, respectively. p. 9. • ) with Pi(t) = Pr{X(t) = i} and P(t) the transition matrix of {X(t)}. p. p. p. (t).!. - ~O I, so that 'p'(t) is indeed a probability distribution vector. 10). ) in the sense of chapter 3. 1. E(t)· AU(P*(t»T fop aZZ t ~ o. PROOF. 7). 12).