This paper is a continuation of a previous paper where we investigated the steady-state behaviour of a stochastic clearing system with Poisson input operated under the following clearing policy : all the quantity is instantaneously removed from the system whenever there are at least M items in the queue, or every t time units since the first arrival after the last clearing, whichever occurs first. This type of policy was termed a bounded M-policy. The objective of this paper is to examine the behaviour of the expected average cost per unit time in the class of bounded M-policies and in the class of T-policies that clear the system every T time units. We find optimal policies in both classes by comparing the associated expected average costs, and present some computational results

MAKIŠ, Viliam

MINE, Hisashi

English

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RIGHT:URL:CITATION:AUTHOR(S):ISSUE DATE:TITLE:Optimal Operating Policies for aStochastic Clearing System withBounded Waiting TimesMINE, Hisashi; MAKI, ViliamMINE, Hisashi ...[et al]. Optimal Operating Policies for a Stochastic Clearing System withBounded Waiting Times. Memoirs of the Faculty of Engineering, Kyoto University 1984,46(4): 1-61984-12-15http://hdl.handle.net/2433/281278Mem. Fae. Eng., Kyoto Univ. Vol. 46, No. 4 (1984) Optimal Operating Policies for a Stochastic Clearing System with Bounded Waiting Times By Hisashi MINE* and Viliam MAKIS** (Received May 9, 1984) Abstract This paper is a continuation of a previous paper where we investigated the steady-state behaviour of a stochastic clearing system with Poisson input operated under the following clearing policy: all the quantity is instantaneously removed from the system whenever there are at least M items in the queue, cir every t time units since the first arrival after the last clearing, whichever occurs first. This type of policy was termed a bounded M-policy. The objective of this paper is to examine the behaviour of the expected average cost per unit time in the class of bounded M-policies and in the class of T-policies that clear the system every T time units. We find optimal policies in both classes by comparing the associated expected average costs, and present some computational results. l. Introduction We consider a stochastic clearing system with Poisson input operated under one of the following service policies: ( i) a T-policy consisting in removing all the quantity currently present in the system every T time units, (ii) a bounded i-policy consisting in clearing whenever there are at least i items in the queue or every T time units since the first arrival after the last clearing, whichever occurs first. In the previous paper8>, we derived the steady state characteristics for the stochastic clearing system operated under a bounded i-policy, and discussed some limiting cases. In this paper, we examine the behaviour of the expected average cost per unit time over an infinite time horizon under each of the above-mentioned service policies. We find optimal policies in both classes, and present some com-putational results. We now consider the following cost structure. The cost of serv-ing i customers is K+ci, where K>O and c is any constant. The waiting cost of i customers in the queue per unit time is h(i) where h( •) is a non-negative function, * Dept. of Applied Marhematics and Physics ** Dept. of Mathematial Methods, Technical University of Transport, Czechoslovakia 2 Hisashi MINE and Viliam MAKIS such that h(i+ 1) -h(i) >r >o (1) for any non-negative integer i. We assume that in both cases the waiting times are bounded and cannot exceed a given constant t>O. To ensure the finitness of the cost incurred in a cycle, we assume that (2) where {X,., u>O) is the input Poisson process with a given intensity J>O. Note that (2) holds for any polynomial or exponential function h( • ). Let h(x) = ea", where a is any real constant. Then En:h(X,.+l)du) = ~h(k+l) 1:q1 (u)du= ~h~) R1(t) +oo ea(l-eak) = ~lqi(t) 1-ea < +oo, where qi(t) = (!?k e-At, Ri(t) = I>q•-1 (u)du, Ro(t) = 1. (3) We proceed to find optimal operating policies in both classes. 2. Optimal Operating Policies In this section, we examine the behavior of the expected average cost per unit time under the service rules defined in· Section 1, and find the optimal policy in each class. First, we investigate case (i). 2. l. The optimal T-policy Consider a clearing policy that dears the system every T<,t time units. The long-run expected average cost can be obtained by applying a renewal argument. (See e.g. Ross4).) We have g(T) = (E(I~h(X,.)du) +K+cE(X7') )!T (4) for any T>O, where H(u)=E(h(X,.)) and {X,., u>O} is the input Poisson process. In the next lemma, we examine the behaviour of the expected average cost g( · ). Lemma 1. Let (1) hold. Then, the function - g( •) is unimodal with mode s0, determined by s0 =linf{T>O: d(T)=O), (5) where d(T)=[udH(u)-K. (6) PrQof, To prove that -g( •) is unimodal with mode s0, it suffices to show that Optimal Operating Policies for a Stochastic Clearing 3 System with Bounded Waiting Times the derivative of g(s) is negative for s<s0 and positive for s>s0• We have for any s>o dg(s) sH(s)-J:H(u)du-K d(s) ~= s2 s2' (7) and from (1), we get dH(s) (+ 00 +00 ) ~= J. t"i h(j)qj_1 (s) - Eo h(j)qj(S) (8) so that d(s) is increasing in s and from (7), we conclude that -g( •) 1s unimodal with mode s0 determined by (5). This completes the proof. The optimal operating policy is given by the following theorem. Theorem 1. Let (1) hold. The optimal clearing time is given by S 0 pt= min (s0, t}, (9) where s0 is determined by (5). Moreover, Sopt¾So¾V2K/(J.y). (10) Proof. Formula (9) follows immediately from Lemma 1. Next, (6) and (8) yield for any s>O J• J.ys2 d(s)>J.y 0udu-K=-2--K, and the result follows, since d(s) is increasing and d(s)<O for s<s0• In the following section, we examine the behaviour of the average cost per unit time in the class of bounded i-policies and again find the optimal policy. 2. 2. The optimal hounded i-policy We assume that the system is operated under a bounded i-policy for some posi-tive integer i, i. e., the system is cleared whenever the queue length reaches level i, or every t time units since the first arrival after the last clearing, whichever occurs first. It was proved elsewhere2> that this type of operating doctrine is the average cost optimal policy for a controlled M/G/1 queue with bounded waiting times. This is a generalization of the model considered here. We proceed to find the expected average cost per unit time. Under any bounded i-policy, the related decision process is a possibly delayed renewal-reward process. The long-run expected average cost per unit time is equal to the expected cost incurred in a cycle divided by the expected length of the cycle. The length of a cycle is the length of time between two successive clearings. The expected length of a cycle was derived in the previous paper8>, and is given by 1 ,_, µ = T 'j;0Rn(t), (11) where R,.(t) is defined by (3). The expected average cost is given by the following 4 Hisashi MINE and Viliam MAKIS theorem. Theorem 2. Under any bounded i-policy, the expected average cost per unit time is of the form (12) Proof. To obtain (12), it suffices to evaluate the expected waiting cost E;(W) incurred in a cycle. (E; denotes the expectation under a bounded i-policy.) Con-ditioning on the number of arrivals Y1 in the time interval (r;i, r;1 + t], we get E;(W) =h(O)/A+ %-tq•(t)E(J:h(Yu+ l)du I Y1=k) + f q•(t)E(fili-,h(Y,.+ l)du I Y1=k), k=l-1 Jo where r;; _ 1 = r;; - r;1 and r;,. is the k-th arrival time in a given cycle. k>i-1 E(flii-,h(Y,.+ l)du I Y1=k) =1ij h(j+ l)E(1Ji+i-1/i I Y1=k) Jo J=O t 1-1 . =k+l;~h(;), (13) We have for (14) where the second equality follows from Theorem 2. 34> and from formula (2.1. 6) 1>. Similarly, E(J:h(Y,.+l)dul Y1=k)=k!l~1h(j), (15) and (13)-(15) yield, after some algebraic manipulations, 1 1-1 • E;(W) =T "{J0h(;)Rj(t), (16) and (12) follows from (11) and (16). This completes the proof. Now we examine the behaviour of the average cost g; given by (12). We denote for any positive integer i 1-1 r(i) = LJ (h(i)-h(n))Rn(t)-AK. '111=0 We have I r(i+ 1) -r(i) = (h(i+ 1) -h(i)) LJ Rn(t), 7'=0 so that r( ·) is increasing. Next, from (12) g;+1 -g;=R;(t)r(i)!(}d Rn(t) ~ R;(t) ), and we get the following theorem. Theorem 3. Let (1) hold. Then g; is unimodal in i with mode i*=inf{i: _,%t (h(i)-h(n))Rn(t)>AK}, (17) (18) and the bounded i*-policy is the optimal operating policy. In the final section, we compare both optimal policies and present some computational results. Optimal Operating Policies for a Stochastic Clearing System with Bounded Waiting Times 3. Comparison of Operating Policies and Concluding Remarks 5 In this section, we assume that h(i) =hi for some h>O and all non-negative integers i. Then H(u)=E(h(X,.)) = J.hu, (' J.hs2 d(s) = J.h J0udu-K= 2 - K, and the optimal clearing time is S 0 pt=min {v2K/ (J.h), t}. (19) Note that for t->+oo, s0;1=v2K/(J.h), which is also the optimal length of time be-tween orders for the inventory system in Sivazlian6 >, and the optimal dispaching time for a two-terminal shuttle system6>. In the class of bounded i-policies, the expected average cost per unit time de-termined by (12) is in the linear case of the form g;=h(u+ ~jRJ(t) )!~Rn(t)+J.c, Table 1. i* and (g,•-lc)/h as functions of J.K/h and ). for t=l. J.K/h J.: 0.2 I 0.6 I 1 I 2 5 I 10 0.5 1, 0.5000 1, 0.5000 1, 0.5000 1, 0.5000 1, 0.5000 1, 0.5000 1 1, 1.0000 1, 1. 0000 1, 1. 0000 1, 1. 0000 1, 1. 0000 1, 1. 0000 1. 5 2, 1. 4233 2, 1. 3445 2, 1. 3063 2, 1. 2681 2, 1. 2508 2, 1. 2500 2 2, 1. 8465 2, 1. 6891 2, 1. 6127 2, 1.5363 2, 1. 5017 2, 1. 5000 2.5 3, 2.2659 3, 2.0310 2, 1. 9190 2, 1. 8044 2, 1. 7525 2, 1. 7500 3 3, 2.6830 3, 2.3489 3, 2.1940 3, 2.0550 3, 2.0023 3, 2.0000 5 5, 4.3500 4, 3. 6113 4, 3.2385 3, 2.8685 3, 2.6796 3, 2.6668 10 9, 8.5167 7, 6,7375 6, 5. 7499 5, 4.6478 5, 4.0495 5, 4.0008 20 17, 16.8500 13, 12.9875 11, 10.7500 8, 7.9999 7, 6.0980 6, 5.8423 30 26, 25. 1833 20, 19.2375 16, 15.7500 12, 11. 3333 8, 7.8762 8, 7.2797 Table 2. (g(s0 p1)-g,•)/h as function of J.K/h and J. for t=l. J.K/h J.: 0.2 0.6 1 2 5 10 0.5 2.1000 0.6333 0.5000 0.5000 0.5000 0.5000 1 4.1000 0.9667 0.5000 0.4142 0.4142 0.4142 1. 5 6.1767 1. 4555 0.6937 0.4639 0.4812 0.4820 2 8.2535 1. 9442 0.8873 0.4637 0.4983 0.5000 2.5 10.3341 2.4356 1. 0810 0.4456 0.4835 0.4861 3 12.4170 2. 9511 1. 3060 0.4450 0.4472 0.4495 5 20.7500 5.0221 2.2615 0.6315 0.4827 0.4955 10 41. 5833 10.2292 4.7501 1.3522 0.4227 0. 4714 20 83.2500 20.6458 9.7500 3.0001 0.4020 0.4823 30 124.9167 31. 0625 14.7500 4.6667 0.6238 0.4663 6 Hisashi M1NE and Viliam MAKIS and the optimal clearing level i* is i*=inf{i: 1~ (i-j)R;(t)>u}, J=O where u = ).K/h. In Table 1, we present pairs i* and (g;•- Ac) /h and in Table 2 the differences (g(s 0p1) - g;•) /h for t= 1 and for several values of u = ).K/h and A. The results in Table 2 reveal that the optimal bounded i-policy is much better than the optimal T-policy. Since bounded i-policies are very simple, they are ideally suited for practical applications and can be used instead of the more traditional policy of a scheduled periodic service. The resulting saving will be substantial especially for the systems with low input intensities. References 1) David, H. A., "Order Statistics", John Wiley and Sons, New York, 1970. 2) Makis, V., "Optimal Control of a Batch Service Queueing System with Bounded Waiting Time", (submitted to Journal of Appl. Prob.). 3) Mine, H., and Makis, V., "The Steady-State Behaviour of a Stochastic Clearing System with Bounded Waiting Times", Memoirs of the Faculty of Engineering, Kyoto University, Vol. 46, Part 3, 1984, pp. 30-36. 4) Ross, S. M., "Applied Probability Models with Optimization Applications", Holden Day, San Francisco, Calif., 1970. 5) Sivazlian, B. D., "A Continuous Review (s, S) Inventory Systems with Arbitrary Inter-arrival Distribution between Unit Demand", Opns. Res., Vol. 22, 1974, pp. 65-71. 6) Weiss, H.J., "Further Results on an Infinite Capacity Shuttle with Control at a Single Terminal", Opns. Res., Vol. 29, 1981, pp.1212-1217. 

Optimal Operating Policies for a Stochastic Clearing System with Bounded Waiting Times

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Optimal Operating Policies for a Stochastic Clearing System with Bounded Waiting Times

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