On the rate of convergence of the asymptotic expansion for the ergodic distribution of a semi-Markov ( s , S ) inventory
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ON THE RATE OF CONVERGENCE OF THE ASYMPTOTIC EXPANSION FOR THE ERGODIC DISTRIBUTION OF A SEMI-MARKOV ( s , S ) INVENTORY MODEL R. T. Aliyeva and Ò. A. Khaniyevb
UDC 218.2
Abstract. A stochastic process is considered that describes the so-called (s, S) inventory model. The asymptotic behavior of the ergodic distribution of the process is investigated. It is established that the ergodic distribution of the process on the interval [s, S] tends to the uniform distribution for sufficiently large values of the parameter b = S - s, and the convergence rate is estimated. Keywords: ( s, S ) inventory model, ergodic distribution, asymptotic expansion, convergence rate.
INTRODUCTION In the present article, a random process X ( t ) is investigated that describes the so-called ( s, S ) inventory (control) model. Numerous works (see, for example, [1–11]) are devoted to this model. The model is investigated with allowance for requirements of inventory theory, queuing theory, reliability theory, actuarial theory, etc.; b = S - s it is well-known as the min-max system and has the following two control parameters: the lower (critical) inventory level s and upper inventory level S. This system functions as follows: random amounts of requests arrive at the system at random moments, and the stock is not replenished if its level exceeds s ; if the existing level is less than or equal to s, then the decision is made to necessarily fill the stock up to the upper level S. To represent the analytical form of the process X ( t ) , describing the ( s, S ) inventory model, we introduce some sequences of random quantities. Let {x n } and {h n }, n ³ 1, be sequences of positive random quantities defined on some probabilistic space (W , J , P ) , and let the quantities within each sequence be independent and equally distributed according to distribution functions F( t ) = P{x 1 £ t} , t > 0 , and F ( x ) = P{h1 £ x} , x > 0 . Let us consider sequences {Tn } and {Y n }, n = 0, 1,... , that form two independent renewal processes n
n
i =1
i =1
Tn = å x i , Y n = å h i , n = 1, 2, K ; T0 = Y 0 = 0 . We also introduce the following integer-valued random quantities that represent the number of jumps of the process X ( t ) before achieving the so-called reference level s: n 0 = 0, n 1 = min {k ³ 1: S - Y k < s},
n n + 1 = min {k ³ n n + 1: S - (Y k - Y n n ) < s}.
Let t n be moments at which the process X ( t ) achieves the reference level s, t n = Tn n , n = 1, 2,K , t 0 = 0. a
Baku State University, Baku, Azerbaijan, [email protected]. bInstitute of Cybernetics, National Academy of Sciences of Azerbaijan, Baku, Azerbaijan and TOBB University of Economics and Technology, Ankara, Turkey, [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 138–143, January–February 2012. Original article submitted October 12, 2009. 1060-0396/12/4801-0117
©
2012 Springer Science+Business Media, Inc.
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We introduce a counting process n( t ) = max{k ³ 0: Tk £ t} that counts the number of jumps of the process X ( t ) on the in
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