PERGAMON Computers & Industrial Engineering 37 (1999) 269-272
oornlmJl:er~ irtduitsr.iml ortsIrml~r.ing
B i l e v e l H y s t e r e t i c S e r v i c e R a t e C o n t r o l F o r B u l k A r r i v a l Q u e u e Fuh-hwa Liu & Jung-wei Tseng
Department of Industrial Engineering & Management National Chiao Tung University
Taiwan, Republic of China ABSTRACT
This paper concems M Ix] / M / 1 queue. The number of customers in each arriving unit is a random variable. There is two control threshold values K and N, K is smaller than N. Service rate is switched from u to tu whenever the system size increases to N. The tu rate is switched to u when the system size drops to the value K. We derive the steady-state probabilities of the number o f customers in system and the expected number of customers in system. A cost model is introduced for the service cost, queuing cost, and switching cost. © 1999 Elsevier Science Ltd. All rights reserved.
Keywords: Queue, Bulk arrival, Service control. I N T R O D U C T I O N
Hysteretic service rate control policy has studied extensively in the literature. The control policy has been studied by number of authors (Gebhard, 1967; Crabill, Gross and Magazine, 1977; Lu and Serfozo, 1984; Teghem, 1986; Gray et al., 1992; Lee et al., 1998; Lin and Kumar, 1984; Wang, 1993). In this paper, we consider M tx] / M / 1 queuing system with unlimit size. The arrival stream forms a Poisson process in which the number o f customers in each arriving unit is a random variable X, with probability density Cx. There is two control threshold values K and N, K is smaller than N. Service rate is switched from u to tu whenever the system size increases to N. The tu rate is switched to u when the system size drops to the value K. We derive the steady- state probabilities of the number of customers in system and the expected number of customers in system. A cost model is introduced for the service cost, queuing cost, and switching cost.
THE MAIN RESULTS
This model may be analyzed by continuous time parameter Markov chain. We divide the state of the system into two classes. F 1 = {(n,1);n = 0 , K , N} be the state in which n customers in system and the service rate is u. F 2 = {(n,2); n = K + 1,K } be the state in which n customers in system and the service rate is tu. Let n(n,1) denotes the steady-state probability of the state (n, 1),
n(n, 2)denotes the steady-state probability of the state (n, 2). n0, rr(0,1) denotes the probability
of empty state (0,1)
The steady-state equations are given as follows:
0 = -Z•(0, 1) + uzr(1, 1), (1)
0 = - ( X + u ) r c ( n , 1)+~-~)Lci~z(n-i , 1 ) + u r r ( n + l , 1 ) , l < n < N , n ; ~ K , (2) i=1
0360-8352/99 - see front matter © 1999 Elsevier Science Ltd. All rights reserved. PII: S0360-8352(99)00071-6
270 Proceedings o f the 24th International Conference on Computers and Industrial Engineering K 0 = - ( , ~ + u ) 1 r ( K , 1)+~-~2~cizr(K-i , 1 ) + t u z r ( K + l , 2 ) + u l r ( K + l , 1), (3) i=1 N 0 = -(7~ + u l r c ( N , 1) + ~ )~c, l r ( N - i, 11, (4/ i = l 0 = -(~. + tu)zr(K + 1, 2) + t u l r ( K + 2, 2), (5) n - K - 1
0 = -(;~ + tu)lr(n, 2) + ~ ~CiT"C(t'l - - i, 2) + tuTr(n + 1, 2), n = K + 2, K , N, (6)
i=1
n - K - I
O = - ( Z + t u ) l r ( n , 2 ) + ~ . c i l r ( n - i , 1)+ Z ~ , c i r c ( n - i , 2 ) + t u l r ( n + l , 2), n = N + I , K , (7)
i=1 i=1
Multiply equations (1)~(7) with appropriate z", and take summation.
~ N 0o
Let C ( z ) = ~ _ c . z " , ~ , ( z ) = }-" ~(i, 1)z i and ~ 2 ( z ) = ~ . , ~ ( i , 2)z i . We obtain,
n=l i = 0 i = K + I
nt (z)[u - (9~ + u ) z + )~zC(z)] + z~ z (z)[tu - (Z + t u ) z + 2zC(z)] = U~ro(1 - z) (8) L e t ~ ( n , 1 ) = ~ . ~ o , l < n < N , and let ~(n, 2 ) = ~ . ~ o , n _ > K + l . Let ~'. (O<_n<_N) be the coefficient of the probability of the empty state in typical M txl / M / 1 queue with service rate u.
P r o p e r t y 1
~r '., n _> 0, that satisfy the following relation tr' o = l , lr[ = $ , where q~ = )~/u i 7r +, = [(1 + j=l P r o p e r t y 2 ft. = r c ' . , O < n < K , zr K÷ i = z~'x÷ i + h i v x+ i, i = 1, K , N - K , Where i - 2 h i = - t , h i = (1 + ~)hi_ I - ~ Z cjhi-j -1, i = 2 , K , N - K , j = l N ~b ~ cix~q_ i - (1 + ¢ ) n n i=l ~gK+l = N - K - 1 (1 + d~)hN_ K - d~ ~ C i h N _ K _ i i=l zr,(1) + zr2 (1) = 1, (9a) (9b) (10a) (10b) (11) that is, rrt(1)+rr2(1 ) = 7r0[~j(1)+~2(1)] (12) Define ~ i ( z ) = T~i(z) / 7~o, i = 1, 2,
By the boundary condition, since Therefore,
lr o = 1 / [n3, (1) + ~2 (1)] Furthermore,
Proceedings of the 24th International Conference on Computers and Industrial Engineering 271
N N N
i=O i=O i=K+l
Divide above equation by ~ o, we obtain,
N N N N - K
7~,(Z) = E I [ ; Z i +lllK+ , E h i _ x z i = E~'cfzi " 4 " ~ K + I E h i Z K + i ,
i=0 i=K+I i=0 i=1
Let S = ~1 (1), we obtain,
N N - K
s= Z,,;
Zh,,
i=0 i=1
Take the first derivative o f equation (13) with respect to z
d N N - K
~ I ( Z ) = EiT[~Z i-! + I//K+ I Z ( K +i)hi zK+i-1 ,
i=l i=l
Let T = d g q dz ~(1), we obtain,
N N - K
T = ~'~ilr; +u/x+l ~'~(K
+i)hi,
i=l i=l
Divide equation (8) by g0, we obtain,
~, (z)[u - (Z + u)z + 2zC(z)] + ~z (z)[tu - (~. + tu)z + 2zC(z)] = u(1 - z)
(13)
(14)
(15)
(16) Take first derivative o f (16) with respect to z and evaluate at z = 1, we obtain,
aq, (1)[-u + ;LE(x)] + ~2 (1)[-tu + 2&:(x)] = - u
Let U = aq2(1 ) , pl = c~E(X), the first traffic intensity. P2 = c~E(X)/t, the second traffic intensity. Assume P2 < 1. We get
U = - u . aq, (1)[-u + EE(X)] _ P2 [-1 + S(1 - p, )
- (17) Pl (P~ - 1) - t u + LE(X) From (12)-(17), we obtain, 1 1 rr° = ~ ( 1 ) + ~ 2 ( 1 ) S + U - P ' ( 1 - P 2 ) (18) P2 + S(Pl - P2)
Take second derivative o f equation (16) with respect to z, and evaluate at z = 1, we obtain, ~z ~2 ( l ) [ - t u + LE(X)] = - 2 ~ z ~3, (1)[-u + 3.E(X)] - {2AJ~(X) + 3 . E [ X ( X - 1)]} [~1 (1) + ~3 2 (1)] 2
Let V= d dz 2 (1), then,
V = 2T(p, - P2) + P2 {2p, + ~ E [ X ( X - 1)]}(S + U)
2p,(1 - P2) (19)
From (12)-(19), we can obtain the expected number o f customers in system L,
L = T ( p , - p ; ) p2{2p, +gpE[X(X-1)]}
+ (20)
272 Proceedings of the 24th International Conference on Computers and Industrial Engineering
SPECIAL CASES
It is interesting that for various combination of (N, K) the model generalize several models. (a) K=N=0, Pr(X=I)=I. The model is reduced to typical M / M / 1 queuing model with service
rate tu.
(b)K=N¢ 0, Pr(X=I)=I. The model reduce to M / M / 1 queuing model with state dependent (c) K=N=0. The model reduce to regular M tx] / M / 1 queuing model with service rate tu
(d)K=N¢ 0. The model is reduced to Mtx} / M / 1 queuing model with state dependent.
(e)Pr(X=l)=l. The model reduce to M~ M / 1 with bilevel hysteretic service rate control (Gebhard, 1967)
REFERENCES
Crabill, T., D. Gross and M. Magazine (1977). A Classified Bibliography of Research on Optimal Design and Control of Queues. Open Res., 25,219-232.
Gebhard, R. (1967). A Queueing Process with Bilevel Hysteretic Service-Rate Control. Naval Res. Logis. Quart., 14, 117-130.
Gray, W., P. Wang, and M. Scott (1992). An M / G / I - T y p e Queueing Model with Service Times Depending on Queue Length. Appl. Math. Model., 16, 652-658.
Lee, H. W., J. G Park, B. K. Kim, S. H. Yoon, B. Y. Ahn, and N. I. Park (1998). Queue Length and Waiting Time Analysis of a Batch Arrival Queue with Bilevel Control. Comput. Open Res., 25, 3, 191-205.
Lin, W. and P. R. Kumar (1984). Optimal Control of a Queueing System with Two Heterogeneous Servers. IEEE Trans. on Automatic Control, 29, 8,696-703.
Lu, F. and R. Serfozo (1984). M / M / 1 Queueing Decision Process with Monotone Hysteretic Optimal Policies. Oper. Res., 32, 5, 1117-1132.
Teghem, J. (1986). Control of the Service Process in a Queueing System. Eun J. Oper. Res.,
23, 145-158.
Wang, P. P. (1993). An M / M / c Type of Queueing Model with (Ri,r,.) Switch-Over Policy.
