IMBEDDED MARKOV CHAIN ANALYSIS OF SINGLE SERVER BULK QUEUES U. NARAYAN BHAT (received 11 August 1963) Summary In this paper results from Fluctuation Theory are used to analyse the imbedded Markov chains of two single server bulk-queueing systems, (i) with Poisson arrivals and arbitrary service time distribution and (ii) with arbitrary inter-arrival time distribution and negative exponential service time. The discrete time transition probabilities and the equiUbrium behaviour of the queue lengths of the systems have been obtained along with distribu­ tions concerning the busy periods. From the general results several special cases have been derived. 0. Introduction The general bulk queue is described as follows: Groups of customers arrive at service points and get served in batches. The sizes of the arriving groups and those of the batches for service are random variables having independent distributions. The time intervals between successive group arrivals are independent and identically distributed random variables; so also are the service times of the different batches. We shall call the maximum size of a service group as the "capacity" for that service and assume that this capacity is independent of the queue length at that time. Following {x) l,l) Kendall [9] we use the notation GI \G \l to represent the general single server bulk queue, the exponents x and y denoting the sizes of the arriving groups and service capacity respectively. We shall suppress these exponents when they are equal to one. Further, we shall assume that the queue-discipline is "first come, first served" and that when the arrivals are in groups, the units will be ordered for the purpose of service. The object of this paper is to obtain the discrete time behaviour of (X) M the bulk queues (i) (Poisson arrivals and arbitrary service W M{V) \G \1 time) and (ii) GI \M \\ (arbitrary inter-arrival time distribution and negative exponential service time). This is done by analysing the Markov chains imbedded in them. Some aspects of these systems have been studied Downloaded from https://www.cambridge.org/core. IP address: 170.106.40.139, on 30 Sep 2021 at 11:07:55, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700023454 [2] Imbedded Markov chain analysis of single server bulk queues 245 by Miller [10]; and several special cases have been considered by Bailey [1], Taiswal [6], Takacs [13,14], Foster [4], Foster and Nyunt [5], Keilson [7] and Boudreau, Griffin and Kac [2]. In our discussion we make use of known results in Fluctuation Theory for the sums of independent and identical random variables, to obtain the behaviour of QN, the queue length at TN (arrival or departure epoch, whichever is convenient). For the study of WN the virtual waiting time at an instant of arrival T„, these results have already been used by Spitzer [12] and Kemperman [8]. The paper is divided into four sections. In section 1 certain basic results from Fluctuation Theory are described; section 2 deals with the system MW\GW\1 and section 3 with the system GIIX)\MIR)\L. Finally some special cases of these queues have been considered in the last section. 1. Basic results from fluctuation theory The following are the special cases of more general results derived by Spitzer [12], Feller [3] and Kemperman [8], for sums of independent and identical random variables (r.v.). Let {ZN} (» = 1, 2 • • •) be a sequence of mutually independent and identical r.v.'s assuming integral values and Sn == Z1+Z2+ • • • -\-ZN (« = 1, 2 • • •), S0 = 0 be the partial suras of {ZN}. Let (1 I) Pr{Z" = i] = *' {j = '} 1 ' ' ¿(0) = £'(0Z"), 0 < f (1) < co and (1.2) *<"> = Pr{SB = 7} («^1), *}« = *„ A<"' = 0 (/#0),*g» = l. We define two functions M~(6, Z) and M+(0, Z) as follows. (1.3) M-(6, Z) = exp I -1 - 2 E'K?]) (W) < l> l0l 2? 1) I 1 « -00 I B) (1.4) M+(D, 2)=exp(2-2^ 1 (WXL 1*1 £1) \ i » o ) such that they are related by the property (1.5) [l-^(0)]M+(0, Z) = M-(6, Z) (Kemperman [8] equations (13.4) —(13.8)). For the partial sums S„, we have the following results, (i) Let g„* = Pr^ > 0, S, > 0 • • • SN_X > 0, SB ^ 0}; then Downloaded from https://www.cambridge.org/core. IP address: 170.106.40.139, on 30 Sep 2021 at 11:07:55, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700023454 246 U. Narayan Bhat [3] (16) g^) = |^=l-exp{-|^|^j = 1-M-(1, z) exp { -1 ~ Pr{SB = 0}}. (ii) Let = Pr{S1> 0, Sa > 0, • • • SB_X > 0, Sn = /} (/ > 0); then n*(B, z) = | | **(?>"ö> = exp (5 - | 0*Ä<">) n=-o /-l t i w i ; (L7) = M+(0, z) exp j -|£ Pr{Sn = 0} j. (üi) Let g„ = Pr^ ^ 0, S2 2t 0 • • • S_t 2i 0, S„ < 0}, then (18) fW-5«^-i— = 1-M-(1, 2). (iv) Let n {i+St i+S ^ j) (i, j n = Pr 2j 0, z 0 • • • *+SB_, 2r 0, i+S„ = Sj 0); 00 CO CO w n then 7t(m, 0, z) =2 2 2 n(*. 7> «W n=0 «=0 ¿«-0 1 /00 »n oo co «n —1 \ (L9) - r^Äex p 2 - 2 ^»+2 - 2 °>-'*H n 1—«0 d M o 1 -oo J ~~ 1—w0M-(w-1, z)" [Spitzer [12]; also Feller [3] equations (9.8), (9.13) and (7.10). For (1.9) here, see Kemperman [8] equation (16.13).] Finally we shall define w S S„); n = max (0, lt S2 • • • then 00 oo / oo .» 0 oo yn 1 = M+(0, z)\M-(l, z)]-1. Downloaded from https://www.cambridge.org/core. IP address: 170.106.40.139, on 30 Sep 2021 at 11:07:55, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700023454 [4] Imbedded Markov chain analysis of single server bulk queues 247 When n -> oo, writing linv,.,^ w„ = , we have (a) if E(Zn) _ 0, »M = oo with probability one; (b) if E(Z„) < 0, tPo, < oo with probability one and is given by 1 (i.ii) £(0»°°) = exp (-2 — I(i-e*)^" ) (Spitzer [12]). (x) 2. The queue yM |a'"|l Description: The queueing system considered here has the following description. (i) The arrivals are in a Poisson process with parameter U in groups of size {CB} having the distribution Pr{CB = r} = cr (r = 0, 1, 2---); let (2.1) c(0) = £(0C«). |0| ^ 1, 0 < c'(l) < oo. The probability that / customers arrive in time interval (0, T) is given by (2.2) aAT) = ^e-XT^lcf) 1 where {cj* } is the ft-fold convolution of {ct} with itself. It should be noted that the compound Poisson process (2.2) has the property that the number of arrivals in non-overlapping time intervals are independent r.v.'s. (ii) The customers are served in batches of variable capacity. Let the successive departures take place at the instants tlt i2, • • •, and denote by vn the service time of the batch departing at tn. We assume that {vn} (« = 1, 2 • • •) is a sequence of identically distributed independent r.v.'s with a common distribution function H(x) = Pr{t>„ ^ a;}. Let (2.3) y>{o) = f e-™dH(x) Re(cr) ^ 0 J o and 0 < — ip'(0) < co. Let Xn be the number of customers arrived during a service period; then we have A c iH (2.4) Pr {Xn = ;} = «, = f° 2 *~ ' '*' ®' Jo H »! (2.5) X and K(d) = E(B ") = v(A-Ac(0)). Downloaded from https://www.cambridge.org/core. IP address: 170.106.40.139, on 30 Sep 2021 at 11:07:55, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700023454 248 U. Narayan Bhat [5] t (n (iii) If Y„ is the capacity for service ending at n+l = 1, 2, 3 • • •), we assume that the r.v.'s Yn are identically distributed and mutually independent and also independent of the Xn; let (2 61 Pr{Yn = /} = &, (/=0,1,2..-); K ' ' B(6) = E(0r'). |0| <i 1, 0 < .B'(l) < oo. The relative traffic intensity of the system is defined by (2.7) gg,),-V(0y(l) (0<o<oo). 1 P ; 1 ; P E(Yn) B'(l) ^ Further, we define Qn = Q(tn+0) where Q(t) is the queue length at time t (number of customers in the system, including those who are being served); {Qn} is a Markov chain imbedded in the process Q(t). Let tn and tn+m be such that Qn-1<Yn_1, QT^YT, {r = n, n+l, • • • n+m—1), Qn+m < Yn+m • During this period (tn, tn+m) full service capacity has been utilized and we shall call such a period "capacity busy period". If at some epoch tr, Qr < Yr, the server is said to be "slack"; then the following different possibilities are open to him at that time: (a) Y„ > Qn 0, he may wait until his maximum capacity Y„ is reached; (b) Y„ > Q„ > 0, he may take the available customers into service; (c) Qn = 0, he may either wait for the first customer to arrive or proceed for service with no customers. An example of the last service mechanism is an elevator or a transport service which is kept in operation even when there are no cus­ tomers to be served.