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A multilevel control bulk queueing system with vacationing server
)perations Research Letters 13 (1993) 183-188 North-Holland
April 1993
A multilevel control bulk queueing system with vacationing server Lev M. Abolnikov Department of Mathematics, Loyola Marymount University, Los Angeles, CA 90045, USA
Jewgeni H. Dshalalow Department of Applied Mathematics, Florida Institute of Technology, Melbourne, FL 32901, USA
Alexander M. Dukhovny Department of Mathematics, San-Francisco State University, San-Francisco, CA 94132, USA Received August 1991 Revised October 1992
This article studies a Markov chain describing the evolution of the queue in a general single-server bulk queueing system with continuously operating (or, equivalently vacationing) server, semi-Markov modulated compound Poisson input, queue length dependent service time, and multilevel control server capacity. We establish a necessary and sufficient criterion for ergodicity of this Markov chain and find its stationary distribution. As an example a single-server queueing system is considered, and some explicit results are obtained in this case. semi-Markov modulated input; controlled group size; controlled service; controlled batch size; single-server queue; vacation; queueing process; embedded Markov chain
1. Introduction I n this a r t i c l e w e study a q u e u e i n g system of type M X / G / 1 with a c o m p o u n d m o d u l a t e d input, state d e p e n d e n t exhaustive service a n d m u l t i p l e vacations. T h e l a t t e r refers to a class o f q u e u e s w h e r e the s e r v e r leaves on v a c a t i o n e a c h t i m e w h e n t h e q u e u e is e x h a u s t e d . T h e server k e e p s on going on v a c a t i o n s as long as t h e system r e m a i n s empty. A v a c a t i o n s e q u e n c e is t e r m i n a t e d w h e n u p o n his r e t u r n t h e server finds at least o n e c u s t o m e r in t h e system. T h i s s i t u a t i o n is e q u i v a l e n t to a so-called c o n t i n u o u s l y o p e r a t i n g server, a t e r m u s e d by t h e a u t h o r s p r e v i o u s l y [3]. In a d d i t i o n , we a s s u m e t h a t the i n p u t s t r e a m o f arriving c u s t o m e r s is m o d u l a t e d by t h e system. T h i s m e a n s t h a t b o t h t h e i n p u t p o i n t p r o c e s s a n d their m a r k s a r e a f f e c t e d by t h e q u e u e i n g p r o c e s s at c e r t a i n decision points, w h e r e p a r a m e t e r s of the p r o c e s s can b e c h a n g e d . B o t h t h e service a n d v a c a t i o n t i m e s also d e p e n d on t h e q u e u e i n g process. T h e system is f o r m a l l y d e s c r i b e d below. L e t {Q(t); t > 0} b e a stochastic p r o c e s s v a l u e d in gt = {0, 1 . . . . } that d e s c r i b e s t h e n u m b e r of units in a s i n g l e - s e r v e r q u e u e i n g system at t i m e t. O t h e r p r o c e s s e s r e l a t e d to Q ( t ) a r e as follows. {tn; n ~ N 0 (t o = 0)}--* ~ + is t h e p o i n t p r o c e s s o f successive m o m e n t s of service c o m p l e t i o n s ; {Qn = Q ( t , + 0); n ~ N 0} --->~ t h e e m b e d d e d p r o c e s s over t h e m o m e n t s o f t i m e {tn}. Input. T o d e s c r i b e t h e i n p u t to t h e system we n e e d t h e n o t i o n of the m o d u l a t i o n of a stochastic p r o c e s s i n t r o d u c e d in D s h a l a l o w [9]. L e t {so(t); t > 0} be an i n t e g e r - v a l u e d j u m p p r o c e s s with successive j u m p s at t , , n ~ N 0 (we allow ~ ( t n ) = ~ ( t n + l ) w i t h positive p r o b a b i l i t y ) a n d let {rk; k e N } b e a 0167-6377/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
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non-stationary orderly Poisson point process with intensity function A(t). Then we call the doubly stochastic Poisson point process with intensity A(~(t)) the Poisson process modulated by the j u m p process ~(t) and it is denoted by ~-~. Denote N¢(.) the associated counting measure. A compound modulated process is defined as follows. Let ( a ) = {al~(t ), o/2¢(t),... } be a doubly stochastic sequence of random variables such that given a fixed value of ~:(t) ( a ) is a sequence of independent identically distributed random variables. Then the compound Poisson modulated process is defined as Z e ( t ) = v.N~(t)~ ~"i= 1 ~i~(t)" Let C(.) be the counting measure associated with the point process tn. Define ~ ( t ) = Q(tc(t)+ 0), v > 0. Then we assume that the input to the system is a compound Poisson process modulated by ~(t) in the light of the above definition with ai6(t) as the ith batch size of the input flow. Thus, in our case ( a ) is an integer-valued doubly stochastic process describing the size of groups of entering units. We denote a~(t)(z) = E[ z~i~t) ], i = 1, 2 , . . . , z ~B(O, 1) U { - 1, 1}, (where B(zo, R ) is an open ball in C centered at z 0 and with radius R), the generating function of ith component of the process (o~). Assume that 0 < ae(t) = E[ai¢(t )] < % i = 1, 2 . . . . . Service time and service discipline. Units, entering the system, are placed in a line in a waiting room with an infinite capacity. Arriving groups of units obey the FIFO discipline. Within a group the order of selecting a unit for service is arbitrary. The servicing facility consists of a single channel that begins a new servicing act even in case if, upon completion of service, he finds no waiting units in the system. It is supposed that the service act is not interrupted by any group of units arriving at the system during this time. We call such a model a queueing system with continuously operating server. Observe that a continuously operating server can also be regarded as a vacationing server. Indeed, the server goes on his first vacation as soon as there are no units in the system at the end of service. The server does not terminate his vacation period even though during this time new units may enter the system. If upon his return to the system there are still no units the server goes on his next vacation, and so on. At the end of each vacation period the server checks if there are waiting units in the system, and if so, he immediately begins a new service act (of a group of units in accordance with the servicing discipline described below). The notion of continuously operating (or vacationing) server is common in the literature on queueing systems (cf. [2,5-8,10-131). It is supposed that at instant t n the server takes for service the nth group of units of the size min{Qn, m(Qn)}, where m ( . ) denotes the capacity of the server (as an integer function with m(0) = 0). The service time o-~ = t~ - t , _ 1 of the nth batch is distributed according to BQ, selected from a given sequences {B0, B1,... } of distribution functions each of which is concentrated on ~+ with b i = Ei[Orl] ~ (0, oo), where E i denotes the expected value with respect to the probability measure pi. Also denote fli(0)= f~ e x p ( - O x ) B i ( d x ) , Re(0) >_ 0. The service times trn are supposed to be conditionally independent given Q,, n = 1, 2 . . . . . In particular, the introduced general state dependence of service assumes that the distribution of vacation periods differs from that of service times. In this paper the embedded process Qn is studied. The authors establish a necessary and sufficient condition for the ergodicity of this process and obtain its stationary distributions in terms of generating functions and roots of a certain associated function in the closed unit disc of the complex plane. The results obtained in the article are illustrated by several examples; one of them (a single-level control bulk queueing system) may be of independent theoretical and practical interest. Some explicit results are obtained in this case. The methods used in the article represent a further development of those first initiated by the authors in [1,2,4,5].
2. Process
Qn
Let vn = Z6(trn), i.e. vn is the number of units arriving at the system during the nth service act given that the queue length at the end of the preceding service was ~. Then by the obvious relation Q~+l = [ Q n - m ( Q ~ )]++ V~+l (where (u) += sup{u, 0}) it follows that {Q~} is a homogeneous Markov chain with the transition probability matrix A = (aij; i, j ~ No). We assume that for some N (which may be arbitrarily large) Ai = A, a j ( z ) = a(z), aj = a, j = N + 1, N + 2 , . . . , m ( N + 1) = m ( N + 2) . . . . .
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m( < N + 1) and BN+ 1 = BN+ 2 . . . . . B. Under these conditions, the transition probability matrix A of the Markov chain {an} is reduced to a so-called Am,N-matrix
A = (aij" i,j ~ rF; aij = kj_i+m, i > N, j >_i - m; aij = O, i > N, j < i - m)
(2.1)
studied earlier by Abolnikov and Dukhovny [4], where the values k i of the corresponding entries aij a r e determined by oo
K(z)=
Ekj zj=fl(A-Aa(z)),
f l ( o ) = fo e x p ( - O x ) B ( d x ) .
j>_o
Let A i ( z ) = FT>_oaijzJ. Then i = 0 , 1 . . . . . N,
Ai(z ) =z(i-m(i))+Ki(z),
Ai(z ) =zi-mK(z),
i=N+I,
N+2 .....
with Ki(z) = fli(Ai - Aiai(z)). The following theorem is the main result of this paper. Theorem. The invariant probability measure P of the Markov Chain Qn with the transition probability Am,N-matrix A exists if and only if p < m, where p = AabN+ v Under this condition, the generating function P ( z ) of the invariant probability measure P satisfies the following relations: N E Pi[zmAi(z)
P ( z ) = i=0
- ziK(z)]
(2.4a)
z m -K(z)
dk ~--0 i Pi~zk [ai(z)-zi]lz=zs=O'
k = 0 " " "' r s - 1, s = 1 '" " " S,
(2.4b)
N
Y'~ Pi[ A~(1) - i + m - p ] = m - p ,
(2.4c)
i=o
where z s are the roots of the function zU+l-m[ zm --K(Z)] in the region B(O, 1)\{1} with the multiplicities Ps, such that F.s~=lr~^ = N. The system of equations (2.4b) and (2.4c) has a unique solution P0,-. ., PN" Proof. Formula (2.4a) follows from P ( z ) = F . i ~ , P i A i ( z ) and (2.2-2.3). It is easy to modify (2.4a) into N
~_~ P i { A i ( z ) - z i}
~_, p zi_(N+l)= i=N+l
'
i=0 zN+I--zN+I-mK(Z)
(2.4d) "
~=N+lPi zi-(N+I) is analytic in B(0, 1) and continuous on the boundary 0B(0, 1). According to theorem 2 [4], the function z ~ z m - K ( z ) must have exactly m zeros (counted with their multiplicities) in B(0, 1), from which all zeros on the boundary 0B(0, 1), including the root 1, must be simple, after we meet the ergodicity condition p < m. Therefore, the denominator on the right hand side of (2.4d) has exactly N roots in the region B(0, 1)\{1} and this, along with (P, 1) = 1 (which is equivalent to (2.4c)), yields (2.4b) and (2.4c). Now we prove the uniqueness of {P0,.-., PN}. Suppose that the system of equations (2.4b) and (2.4c) has another solution p * = {Pi*; i = 0 . . . . . N}. We substitute p* into (2.4a) to obtain the generating function P*(z). Then, P * ( z ) is analytic in B(0, 1) and continuous on 0B(0, 1). Therefore, P* = {pz*; i ~ a/,} ~ (11, II" II). Obviously, P * ( z ) = E i ~ p i * A i ( z ) and Obviously,
P*(z) =
Y~Ni=oPi_* { Z m A i ( Z ) -- ziK ( z)} Z m
-K(z) 185
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are equivalent. The last equation is also equivalent to P * = P*A. Since p * satisfies (2.4c) it follows that ( P * , 1 ) = 1. Thus, the system of equations x =xA, (x, 1 ) = 1 has two different solutions in (l x, I1" II) which is impossible. []
3. Examples (i) For
rn(j)= 1, j = 1, 2 . . . . . (2.4a) turns into N
PoKo(z)(1 - z ) + E Pixi(K(z) - K i ( z ) ) j=0
P(z) =
(3.1)
K(z) - z
that further reduces to the Kendall's formula when no control is assumed over both the input and service processes. Observe that the Kendall's formula (established for classical single-server queueing systems with idle periods of the server) holds for the embedded process in our model with vacationing server. This phenomenon is due to the effect of exponentially distributed idleness in the classical model. However, it can be shown [3] that this does not take place for the process with continuous time parameter. (ii) As an illustration to the above formulas let us consider a single-level control bulk model under the following condition. Assume that if the number of units in the system does not exceed N when the server starts its service, then the server capacity is l = m ( 1 ) = m(2) . . . . . m(N), the service time is distributed in accordance with the distribution function B 0 = B l . . . . Bu and )t o = )tl . . . . . )tN" If the number of units in the system is at least N + 1 when the server starts servicing a group of units, then the server capacity is m and the service time is distributed in accordance with the distribution function B. We assume that l < m. Under this restriction, (2.4a) for the generating function P(z) is reduced to 1-1
zm-lKo(Z) F.,p
N
(x
-zO + [Ko(z)zm-t-K(z)] F_,pjz
j=O
e(z) =
j=O z m- K ( z )
,
(3.2)
where P0, Pl . . . . , PN can be derived from (2.4b) and (2.4c). However, in this special case it turns out that an alternative approach can be employed which has an obvious advantage over the general method. Since we can only use m conditions (or equations) to derive m unknown probabilities (of the total N + l) based on m zeros of the function z m - K ( z ) , we will show the way how to find the other N + 1 - m probabilities. First observe that for l = m, B 0 = B (i.e. K 0 = K ) and )to = )t, (3.2) turns into
I-1 K ( z ) ( z l _ z j) P ( z ) = E P j zZ_K(z)
(3.3)
j=0
giving an expression for the generating function /~ in the model with no control. Comparing the corresponding transition Am,~-matrices in the one-level control system and the system without control and following T h e o r e m 2 in Abolnikov [1] we express the first N - m + l + 1 probabilities in terms of the generating function /5(z) in the following form E i=o N-m+t pyz j= (TN_m+lISXz), w h e r e the operator T~, applied to a Taylor series, gives its k th truncation. Taking the latter into account and substituting (3.3) into (3.2) we obtain the expression for the generating function P ( z ) : e(z)---
/1(
( j~=Op1 ( Zm - Zm-I+J) K°( z ) "b [ K°( z ) zm-I - K ( z ) ] TN-m+t K°( z )(Ko(--~ zt -
+[Ko(z)zm-'-K(z)]
E
j=N-m+l+l
)
p zJ (3.4)
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reducing the n u m b e r o f u n k n o w n probabilities to rn. As it follows from the t h e o r e m in Section 2, all of t h e m can be o b t a i n e d in terms of the roots of z m - K ( z ) in the closed unit ball B(0, 1), since P ( z ) is analytic in B(0, 1) and continuous on 0B(0, 1). In particular, for l = m = 1 we obtain f r o m (3.4)
( P(z)=po
" Ko(z)(z-1)+[Ko(z
)-K(z)]T
K°(z)(z-1)) N zS-K~z~
[z-K(z)]-l
with
1 -P0
Po =
go(z) 1 - (Po--P)DN
,
z -Ko(z )
w h e r e o p e r a t o r D i is defined as
1
di
O,& = ~ lim° --~x~&(x).
(3.5)
(iii) As a special case of (ii), consider single-level control bulk model with l = N. H e r e in fact the server takes for service the whole line if the n u m b e r of units does not exceed level N which means: min{j, rn(j)} = j , j = 1 , . . . , N. Since A i ( z ) = Ko(z), i < N, from (2.4b) we have N
N
Y'~pizJ=CI~o(Z),
c = Y'~pj,
j=O
(3.6)
j=O
w h e r e / ~ 0 ( z ) is a N t h d e g r e e interpolating polynomial of Ko(z) taking on the values of Ko(z) and all its derivatives up to (r s - 1)-st o r d e r at z = z,, s = 1 , . . . , S, and at z = 1. Observe that, since zero root is one of z s with multiplicity N + 1 - m , the first N + 1 - m coefficients o f / ~ 0 are as follows:
Dil~ o = D i K o = k} °) (the i-th term of the series Ko( z ) ), i = O, 1 . . . . . N - m, w h e r e o p e r a t o r D i is defined in (3.5). In terms of the notation in (3.6), formula (2.4a) can be rewritten as
- K ( z ) I~o( Z ) z m -K(z)
(3.7)
c = ( m - p ) ( m - p + Po - / ~ ; ( 1 ) ) - '
(3.8)
zmKo(
Z)
P(z) = c with
by (2.4c). N o t e that due to (3.6) Pi = ck} °), k = 0, 1 , . . . , N - m. Consider the above results for rn = 1. In this case
i~o(Z) = ~_, k}O)zj+zN y" ,.(o) r~j , j=0
C
= ( 1 -- p) 1 -- P + E
j=N
(J-N)k}
°)
j=N o~
Pi = ck} °), i = O, 1 . . . . . N - 1, PN = C Y'~ k} °). j=N
A n o t h e r special case is for l = N = 1 and m = 2. T h e n /~0(z) = 1 + (z - 1)[K0(z 1) - l](z 1 - 1) - l , where z I is the only root of z 2 - K ( z ) in B(0, 1). Similar calculations yield
c=(2_p)[Z_p+po_(Ko(zx)_l)(z Pl =C[Ko(Z1)
-
_ 1 ) - , ] -1, P o = C [ z l _ K o ( z l ) ] ( z l _ l
) 1,
1 ] ( z , - 1) - 1 . 187
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(iv) Finally, c o n s i d e r t h e m o d e l with n o c o n t r o l over service d u r a t i o n s a n d o n e level c o n t r o l o f service capacity. T h u s we a s s u m e (as at t h e b e g i n n i n g o f E x a m p l e (iii)) t h a t t h e service c a p a c i t y switches f r o m N to m at level N (of t h e n u m b e r o f units in t h e system), t h e r e b y r e d u c i n g (3.7) a n d (3.8) to P(z)
.
cK(z)[zm-I~(z)] . . z'-K(z)
,
.
c
(m
p)(m
^' K (1))
1
1
.
Acknowledgement T h e a u t h o r s a r e t h a n k f u l to t h e r e f e r e e for his critical c o m m e n t s a n d v a l u a b l e r e m a r k s .
References [1] L.M. Abolnikov, "Investigation of a class of discrete Markov processes", lzv. Akad. Nauk SSSR, Techn. Kybemet 15/2, 69-82 [Engl. Transl. Engm. Cybern. 15/2, 51-63 (1977).] [2] L.M. Abolnikov, "A single-level control in Moran's problem with feedback", Oper. Res. Lett. 2/1, 16-19 (1983). [3] L.H. Abolnikov, J. Dshalalow and A. Dukhovny, "A multilevel control bulk queueing system with modulated input and continuously operating server" continuous time parameter queueing process", Tech. Rep. Florida Institute of Technology, MA 089104, 1991. [4] L.M. Abolnikov and A.M. Dukhovny, "Markov chains with transition delta-matrix: Ergodicity conditions, invariant probability measure and applications", J. Agpl. Math. Stoch. Anal. 4/4, 335-355 (1991). [5] L.M. Abolnikov and M. Ya. Postan, "On duality relationship in queueing systems with group arrivals, group service and feedback", Izv. Nauk SSSR, Tech. Kybemet 19/1, 78-85 (1980) [Engl. Transl., Engin. Cybern.19/1, 64-71 (1980)]. [6] N.T.J. Bailey, "On queueing processes with bulk service", J. Royal Star. Soc. Ser. B 16, 83-87 (1954). [7] M.L. Chaudhry and J.G.C. Templeton, A First Course in Bulk Queues, John Wiley and Sons, New York, 1983. [8] B.T. Doshi, "Single server queues with vacations", in: H. Takagi (ed.), Stochastic Analysis of Computer and Communication Systems, Elsevier Science Publishers B.V., North-Holland, 1990. [9] J. Dshalalow, "On modulated random measures", J. Appl. Math. Stoch. Anal. 4/4, 305-312 (1991). [10] N.K. Jaiswal, "A bulk queueing problem with variable capacity", J. Royal Star. Soc. Ser. B 23, 143-148 (1961). [11] N.S. Kambo and M.L. Chaudhry, "On two bulk-service queueing models", J. Inform. Optim. Sci. 5/3, 279-292 (1984). [12] I.M. Stuart and G.B. McMahon, "A queueing system with bulk service", Oper. Res. 14, 728-831 (1966). [13] H. Takagi, QueueingAnalysis, Vol. 1: Vacation and Priority Systems, Elsevier Science Publishers B.V. North-Holland, 1991.
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A multilevel control bulk queueing system with vacationing server Lev M. Abolnikov Department of Mathematics, Loyola Marymount University, Los Angeles, CA 90045, USA
Jewgeni H. Dshalalow Department of Applied Mathematics, Florida Institute of Technology, Melbourne, FL 32901, USA
Alexander M. Dukhovny Department of Mathematics, San-Francisco State University, San-Francisco, CA 94132, USA Received August 1991 Revised October 1992
This article studies a Markov chain describing the evolution of the queue in a general single-server bulk queueing system with continuously operating (or, equivalently vacationing) server, semi-Markov modulated compound Poisson input, queue length dependent service time, and multilevel control server capacity. We establish a necessary and sufficient criterion for ergodicity of this Markov chain and find its stationary distribution. As an example a single-server queueing system is considered, and some explicit results are obtained in this case. semi-Markov modulated input; controlled group size; controlled service; controlled batch size; single-server queue; vacation; queueing process; embedded Markov chain
1. Introduction I n this a r t i c l e w e study a q u e u e i n g system of type M X / G / 1 with a c o m p o u n d m o d u l a t e d input, state d e p e n d e n t exhaustive service a n d m u l t i p l e vacations. T h e l a t t e r refers to a class o f q u e u e s w h e r e the s e r v e r leaves on v a c a t i o n e a c h t i m e w h e n t h e q u e u e is e x h a u s t e d . T h e server k e e p s on going on v a c a t i o n s as long as t h e system r e m a i n s empty. A v a c a t i o n s e q u e n c e is t e r m i n a t e d w h e n u p o n his r e t u r n t h e server finds at least o n e c u s t o m e r in t h e system. T h i s s i t u a t i o n is e q u i v a l e n t to a so-called c o n t i n u o u s l y o p e r a t i n g server, a t e r m u s e d by t h e a u t h o r s p r e v i o u s l y [3]. In a d d i t i o n , we a s s u m e t h a t the i n p u t s t r e a m o f arriving c u s t o m e r s is m o d u l a t e d by t h e system. T h i s m e a n s t h a t b o t h t h e i n p u t p o i n t p r o c e s s a n d their m a r k s a r e a f f e c t e d by t h e q u e u e i n g p r o c e s s at c e r t a i n decision points, w h e r e p a r a m e t e r s of the p r o c e s s can b e c h a n g e d . B o t h t h e service a n d v a c a t i o n t i m e s also d e p e n d on t h e q u e u e i n g process. T h e system is f o r m a l l y d e s c r i b e d below. L e t {Q(t); t > 0} b e a stochastic p r o c e s s v a l u e d in gt = {0, 1 . . . . } that d e s c r i b e s t h e n u m b e r of units in a s i n g l e - s e r v e r q u e u e i n g system at t i m e t. O t h e r p r o c e s s e s r e l a t e d to Q ( t ) a r e as follows. {tn; n ~ N 0 (t o = 0)}--* ~ + is t h e p o i n t p r o c e s s o f successive m o m e n t s of service c o m p l e t i o n s ; {Qn = Q ( t , + 0); n ~ N 0} --->~ t h e e m b e d d e d p r o c e s s over t h e m o m e n t s o f t i m e {tn}. Input. T o d e s c r i b e t h e i n p u t to t h e system we n e e d t h e n o t i o n of the m o d u l a t i o n of a stochastic p r o c e s s i n t r o d u c e d in D s h a l a l o w [9]. L e t {so(t); t > 0} be an i n t e g e r - v a l u e d j u m p p r o c e s s with successive j u m p s at t , , n ~ N 0 (we allow ~ ( t n ) = ~ ( t n + l ) w i t h positive p r o b a b i l i t y ) a n d let {rk; k e N } b e a 0167-6377/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
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non-stationary orderly Poisson point process with intensity function A(t). Then we call the doubly stochastic Poisson point process with intensity A(~(t)) the Poisson process modulated by the j u m p process ~(t) and it is denoted by ~-~. Denote N¢(.) the associated counting measure. A compound modulated process is defined as follows. Let ( a ) = {al~(t ), o/2¢(t),... } be a doubly stochastic sequence of random variables such that given a fixed value of ~:(t) ( a ) is a sequence of independent identically distributed random variables. Then the compound Poisson modulated process is defined as Z e ( t ) = v.N~(t)~ ~"i= 1 ~i~(t)" Let C(.) be the counting measure associated with the point process tn. Define ~ ( t ) = Q(tc(t)+ 0), v > 0. Then we assume that the input to the system is a compound Poisson process modulated by ~(t) in the light of the above definition with ai6(t) as the ith batch size of the input flow. Thus, in our case ( a ) is an integer-valued doubly stochastic process describing the size of groups of entering units. We denote a~(t)(z) = E[ z~i~t) ], i = 1, 2 , . . . , z ~B(O, 1) U { - 1, 1}, (where B(zo, R ) is an open ball in C centered at z 0 and with radius R), the generating function of ith component of the process (o~). Assume that 0 < ae(t) = E[ai¢(t )] < % i = 1, 2 . . . . . Service time and service discipline. Units, entering the system, are placed in a line in a waiting room with an infinite capacity. Arriving groups of units obey the FIFO discipline. Within a group the order of selecting a unit for service is arbitrary. The servicing facility consists of a single channel that begins a new servicing act even in case if, upon completion of service, he finds no waiting units in the system. It is supposed that the service act is not interrupted by any group of units arriving at the system during this time. We call such a model a queueing system with continuously operating server. Observe that a continuously operating server can also be regarded as a vacationing server. Indeed, the server goes on his first vacation as soon as there are no units in the system at the end of service. The server does not terminate his vacation period even though during this time new units may enter the system. If upon his return to the system there are still no units the server goes on his next vacation, and so on. At the end of each vacation period the server checks if there are waiting units in the system, and if so, he immediately begins a new service act (of a group of units in accordance with the servicing discipline described below). The notion of continuously operating (or vacationing) server is common in the literature on queueing systems (cf. [2,5-8,10-131). It is supposed that at instant t n the server takes for service the nth group of units of the size min{Qn, m(Qn)}, where m ( . ) denotes the capacity of the server (as an integer function with m(0) = 0). The service time o-~ = t~ - t , _ 1 of the nth batch is distributed according to BQ, selected from a given sequences {B0, B1,... } of distribution functions each of which is concentrated on ~+ with b i = Ei[Orl] ~ (0, oo), where E i denotes the expected value with respect to the probability measure pi. Also denote fli(0)= f~ e x p ( - O x ) B i ( d x ) , Re(0) >_ 0. The service times trn are supposed to be conditionally independent given Q,, n = 1, 2 . . . . . In particular, the introduced general state dependence of service assumes that the distribution of vacation periods differs from that of service times. In this paper the embedded process Qn is studied. The authors establish a necessary and sufficient condition for the ergodicity of this process and obtain its stationary distributions in terms of generating functions and roots of a certain associated function in the closed unit disc of the complex plane. The results obtained in the article are illustrated by several examples; one of them (a single-level control bulk queueing system) may be of independent theoretical and practical interest. Some explicit results are obtained in this case. The methods used in the article represent a further development of those first initiated by the authors in [1,2,4,5].
2. Process
Qn
Let vn = Z6(trn), i.e. vn is the number of units arriving at the system during the nth service act given that the queue length at the end of the preceding service was ~. Then by the obvious relation Q~+l = [ Q n - m ( Q ~ )]++ V~+l (where (u) += sup{u, 0}) it follows that {Q~} is a homogeneous Markov chain with the transition probability matrix A = (aij; i, j ~ No). We assume that for some N (which may be arbitrarily large) Ai = A, a j ( z ) = a(z), aj = a, j = N + 1, N + 2 , . . . , m ( N + 1) = m ( N + 2) . . . . .
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m( < N + 1) and BN+ 1 = BN+ 2 . . . . . B. Under these conditions, the transition probability matrix A of the Markov chain {an} is reduced to a so-called Am,N-matrix
A = (aij" i,j ~ rF; aij = kj_i+m, i > N, j >_i - m; aij = O, i > N, j < i - m)
(2.1)
studied earlier by Abolnikov and Dukhovny [4], where the values k i of the corresponding entries aij a r e determined by oo
K(z)=
Ekj zj=fl(A-Aa(z)),
f l ( o ) = fo e x p ( - O x ) B ( d x ) .
j>_o
Let A i ( z ) = FT>_oaijzJ. Then i = 0 , 1 . . . . . N,
Ai(z ) =z(i-m(i))+Ki(z),
Ai(z ) =zi-mK(z),
i=N+I,
N+2 .....
with Ki(z) = fli(Ai - Aiai(z)). The following theorem is the main result of this paper. Theorem. The invariant probability measure P of the Markov Chain Qn with the transition probability Am,N-matrix A exists if and only if p < m, where p = AabN+ v Under this condition, the generating function P ( z ) of the invariant probability measure P satisfies the following relations: N E Pi[zmAi(z)
P ( z ) = i=0
- ziK(z)]
(2.4a)
z m -K(z)
dk ~--0 i Pi~zk [ai(z)-zi]lz=zs=O'
k = 0 " " "' r s - 1, s = 1 '" " " S,
(2.4b)
N
Y'~ Pi[ A~(1) - i + m - p ] = m - p ,
(2.4c)
i=o
where z s are the roots of the function zU+l-m[ zm --K(Z)] in the region B(O, 1)\{1} with the multiplicities Ps, such that F.s~=lr~^ = N. The system of equations (2.4b) and (2.4c) has a unique solution P0,-. ., PN" Proof. Formula (2.4a) follows from P ( z ) = F . i ~ , P i A i ( z ) and (2.2-2.3). It is easy to modify (2.4a) into N
~_~ P i { A i ( z ) - z i}
~_, p zi_(N+l)= i=N+l
'
i=0 zN+I--zN+I-mK(Z)
(2.4d) "
~=N+lPi zi-(N+I) is analytic in B(0, 1) and continuous on the boundary 0B(0, 1). According to theorem 2 [4], the function z ~ z m - K ( z ) must have exactly m zeros (counted with their multiplicities) in B(0, 1), from which all zeros on the boundary 0B(0, 1), including the root 1, must be simple, after we meet the ergodicity condition p < m. Therefore, the denominator on the right hand side of (2.4d) has exactly N roots in the region B(0, 1)\{1} and this, along with (P, 1) = 1 (which is equivalent to (2.4c)), yields (2.4b) and (2.4c). Now we prove the uniqueness of {P0,.-., PN}. Suppose that the system of equations (2.4b) and (2.4c) has another solution p * = {Pi*; i = 0 . . . . . N}. We substitute p* into (2.4a) to obtain the generating function P*(z). Then, P * ( z ) is analytic in B(0, 1) and continuous on 0B(0, 1). Therefore, P* = {pz*; i ~ a/,} ~ (11, II" II). Obviously, P * ( z ) = E i ~ p i * A i ( z ) and Obviously,
P*(z) =
Y~Ni=oPi_* { Z m A i ( Z ) -- ziK ( z)} Z m
-K(z) 185
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are equivalent. The last equation is also equivalent to P * = P*A. Since p * satisfies (2.4c) it follows that ( P * , 1 ) = 1. Thus, the system of equations x =xA, (x, 1 ) = 1 has two different solutions in (l x, I1" II) which is impossible. []
3. Examples (i) For
rn(j)= 1, j = 1, 2 . . . . . (2.4a) turns into N
PoKo(z)(1 - z ) + E Pixi(K(z) - K i ( z ) ) j=0
P(z) =
(3.1)
K(z) - z
that further reduces to the Kendall's formula when no control is assumed over both the input and service processes. Observe that the Kendall's formula (established for classical single-server queueing systems with idle periods of the server) holds for the embedded process in our model with vacationing server. This phenomenon is due to the effect of exponentially distributed idleness in the classical model. However, it can be shown [3] that this does not take place for the process with continuous time parameter. (ii) As an illustration to the above formulas let us consider a single-level control bulk model under the following condition. Assume that if the number of units in the system does not exceed N when the server starts its service, then the server capacity is l = m ( 1 ) = m(2) . . . . . m(N), the service time is distributed in accordance with the distribution function B 0 = B l . . . . Bu and )t o = )tl . . . . . )tN" If the number of units in the system is at least N + 1 when the server starts servicing a group of units, then the server capacity is m and the service time is distributed in accordance with the distribution function B. We assume that l < m. Under this restriction, (2.4a) for the generating function P(z) is reduced to 1-1
zm-lKo(Z) F.,p
N
(x
-zO + [Ko(z)zm-t-K(z)] F_,pjz
j=O
e(z) =
j=O z m- K ( z )
,
(3.2)
where P0, Pl . . . . , PN can be derived from (2.4b) and (2.4c). However, in this special case it turns out that an alternative approach can be employed which has an obvious advantage over the general method. Since we can only use m conditions (or equations) to derive m unknown probabilities (of the total N + l) based on m zeros of the function z m - K ( z ) , we will show the way how to find the other N + 1 - m probabilities. First observe that for l = m, B 0 = B (i.e. K 0 = K ) and )to = )t, (3.2) turns into
I-1 K ( z ) ( z l _ z j) P ( z ) = E P j zZ_K(z)
(3.3)
j=0
giving an expression for the generating function /~ in the model with no control. Comparing the corresponding transition Am,~-matrices in the one-level control system and the system without control and following T h e o r e m 2 in Abolnikov [1] we express the first N - m + l + 1 probabilities in terms of the generating function /5(z) in the following form E i=o N-m+t pyz j= (TN_m+lISXz), w h e r e the operator T~, applied to a Taylor series, gives its k th truncation. Taking the latter into account and substituting (3.3) into (3.2) we obtain the expression for the generating function P ( z ) : e(z)---
/1(
( j~=Op1 ( Zm - Zm-I+J) K°( z ) "b [ K°( z ) zm-I - K ( z ) ] TN-m+t K°( z )(Ko(--~ zt -
+[Ko(z)zm-'-K(z)]
E
j=N-m+l+l
)
p zJ (3.4)
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reducing the n u m b e r o f u n k n o w n probabilities to rn. As it follows from the t h e o r e m in Section 2, all of t h e m can be o b t a i n e d in terms of the roots of z m - K ( z ) in the closed unit ball B(0, 1), since P ( z ) is analytic in B(0, 1) and continuous on 0B(0, 1). In particular, for l = m = 1 we obtain f r o m (3.4)
( P(z)=po
" Ko(z)(z-1)+[Ko(z
)-K(z)]T
K°(z)(z-1)) N zS-K~z~
[z-K(z)]-l
with
1 -P0
Po =
go(z) 1 - (Po--P)DN
,
z -Ko(z )
w h e r e o p e r a t o r D i is defined as
1
di
O,& = ~ lim° --~x~&(x).
(3.5)
(iii) As a special case of (ii), consider single-level control bulk model with l = N. H e r e in fact the server takes for service the whole line if the n u m b e r of units does not exceed level N which means: min{j, rn(j)} = j , j = 1 , . . . , N. Since A i ( z ) = Ko(z), i < N, from (2.4b) we have N
N
Y'~pizJ=CI~o(Z),
c = Y'~pj,
j=O
(3.6)
j=O
w h e r e / ~ 0 ( z ) is a N t h d e g r e e interpolating polynomial of Ko(z) taking on the values of Ko(z) and all its derivatives up to (r s - 1)-st o r d e r at z = z,, s = 1 , . . . , S, and at z = 1. Observe that, since zero root is one of z s with multiplicity N + 1 - m , the first N + 1 - m coefficients o f / ~ 0 are as follows:
Dil~ o = D i K o = k} °) (the i-th term of the series Ko( z ) ), i = O, 1 . . . . . N - m, w h e r e o p e r a t o r D i is defined in (3.5). In terms of the notation in (3.6), formula (2.4a) can be rewritten as
- K ( z ) I~o( Z ) z m -K(z)
(3.7)
c = ( m - p ) ( m - p + Po - / ~ ; ( 1 ) ) - '
(3.8)
zmKo(
Z)
P(z) = c with
by (2.4c). N o t e that due to (3.6) Pi = ck} °), k = 0, 1 , . . . , N - m. Consider the above results for rn = 1. In this case
i~o(Z) = ~_, k}O)zj+zN y" ,.(o) r~j , j=0
C
= ( 1 -- p) 1 -- P + E
j=N
(J-N)k}
°)
j=N o~
Pi = ck} °), i = O, 1 . . . . . N - 1, PN = C Y'~ k} °). j=N
A n o t h e r special case is for l = N = 1 and m = 2. T h e n /~0(z) = 1 + (z - 1)[K0(z 1) - l](z 1 - 1) - l , where z I is the only root of z 2 - K ( z ) in B(0, 1). Similar calculations yield
c=(2_p)[Z_p+po_(Ko(zx)_l)(z Pl =C[Ko(Z1)
-
_ 1 ) - , ] -1, P o = C [ z l _ K o ( z l ) ] ( z l _ l
) 1,
1 ] ( z , - 1) - 1 . 187
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(iv) Finally, c o n s i d e r t h e m o d e l with n o c o n t r o l over service d u r a t i o n s a n d o n e level c o n t r o l o f service capacity. T h u s we a s s u m e (as at t h e b e g i n n i n g o f E x a m p l e (iii)) t h a t t h e service c a p a c i t y switches f r o m N to m at level N (of t h e n u m b e r o f units in t h e system), t h e r e b y r e d u c i n g (3.7) a n d (3.8) to P(z)
.
cK(z)[zm-I~(z)] . . z'-K(z)
,
.
c
(m
p)(m
^' K (1))
1
1
.
Acknowledgement T h e a u t h o r s a r e t h a n k f u l to t h e r e f e r e e for his critical c o m m e n t s a n d v a l u a b l e r e m a r k s .
References [1] L.M. Abolnikov, "Investigation of a class of discrete Markov processes", lzv. Akad. Nauk SSSR, Techn. Kybemet 15/2, 69-82 [Engl. Transl. Engm. Cybern. 15/2, 51-63 (1977).] [2] L.M. Abolnikov, "A single-level control in Moran's problem with feedback", Oper. Res. Lett. 2/1, 16-19 (1983). [3] L.H. Abolnikov, J. Dshalalow and A. Dukhovny, "A multilevel control bulk queueing system with modulated input and continuously operating server" continuous time parameter queueing process", Tech. Rep. Florida Institute of Technology, MA 089104, 1991. [4] L.M. Abolnikov and A.M. Dukhovny, "Markov chains with transition delta-matrix: Ergodicity conditions, invariant probability measure and applications", J. Agpl. Math. Stoch. Anal. 4/4, 335-355 (1991). [5] L.M. Abolnikov and M. Ya. Postan, "On duality relationship in queueing systems with group arrivals, group service and feedback", Izv. Nauk SSSR, Tech. Kybemet 19/1, 78-85 (1980) [Engl. Transl., Engin. Cybern.19/1, 64-71 (1980)]. [6] N.T.J. Bailey, "On queueing processes with bulk service", J. Royal Star. Soc. Ser. B 16, 83-87 (1954). [7] M.L. Chaudhry and J.G.C. Templeton, A First Course in Bulk Queues, John Wiley and Sons, New York, 1983. [8] B.T. Doshi, "Single server queues with vacations", in: H. Takagi (ed.), Stochastic Analysis of Computer and Communication Systems, Elsevier Science Publishers B.V., North-Holland, 1990. [9] J. Dshalalow, "On modulated random measures", J. Appl. Math. Stoch. Anal. 4/4, 305-312 (1991). [10] N.K. Jaiswal, "A bulk queueing problem with variable capacity", J. Royal Star. Soc. Ser. B 23, 143-148 (1961). [11] N.S. Kambo and M.L. Chaudhry, "On two bulk-service queueing models", J. Inform. Optim. Sci. 5/3, 279-292 (1984). [12] I.M. Stuart and G.B. McMahon, "A queueing system with bulk service", Oper. Res. 14, 728-831 (1966). [13] H. Takagi, QueueingAnalysis, Vol. 1: Vacation and Priority Systems, Elsevier Science Publishers B.V. North-Holland, 1991.
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