ON APPLICATIONS OF LITTLE ’ S FORMULA

In classical Little’s formula L = AW used in queueing, the parameter A serves as the intensity of the input process. In various applications this parameter may not be known. The author discusses important classes of modulated input processes, where this parameter can be found.


INTRODUCTION
Little's formula, L = AW, belongs to one of the most distinguished and widely referred results in queueing.This formula connects the expected value, L, of the queue length and, W, waiting time processes (in their stochastic equilibrium) through A, the total intensity of the input process.The total intensity of the input process is defined as the rate of incoming customers averaged over the infinite time horizon.More formally: let -{T,,X} be a marked point process (i.e. .Ag = ,= 1Xisi where Sk denotes the unit mass at point k).The total intensity of is defined as A-/t/rnoE[atg([0,t])].Specifically, for a Poisson input process with parameter A the total intensity coincides with this parameter, and for general recurrent-positive renewal processes A is the reciprocal of the mean inter-renewal time.
In many standard queueing systems studied in the past the total intensity was supposed to be either known as an input parameter or it could be obtained analytically, primarily for renewal processes.For various other processes the search for A can become a separate problem.Needless to say, the lack of information on A 1Received" September, 1992.Revised: March, 1993.makes Litle's formula impractical, since i is essential in evaluating W with L avMlable or visa vers.
While i is virtually impossible o give a recipe for A in each case, we discuss some applications o queueing in he class of "modulated inpu processes."The mos general scenario of such a process would be he one of a bulk arrival sream, formally represented as a marked random measure ./tt= i= 1Xicri perturbed by an arbitra- ry (separable) jump process changing its values at random instants of time To, Tx, The modulation assumes that over intervals (T, T + ) input will evolve "by itself' with no affect of , but at each time point T,, characteristics (such as pro- bability distributions) of both ri and Xi will depend on the value takes at T. Markov-modulated Poisson processes form a very important special class of modulated processes arising in telecommunications and queueing [4].In general, a Markov-modulated Poisson process is characterized by a pair of a nonstationry Pois- son process P with intensity/k(t) and a jump Markov process that makes this in- tensity A(Mt(t)).[An excellent survey of Markov-modulated Poisson processes and their applications to queueing may be found in Prabhu and Ztlu [3]].
More general than Markov modulated Poisson processes are semi-Markov and semi-regenerative modulated Poisson process.They find a large variety of applica- tions in networking, inventories, queuing, dams and stock markets.
In this paper the author restricts to an extension and discussion of some re- sults from his earlier paper [2] on intensities for semi-Markov and semi-regenerative modulated processes and their applications to Litle's formula.

BACKGROUND
The following definitions and results are due to Dshalalow [2].
Let {gt, zS, P,((t), tg} (q,()) be a stochastic jump process on g (where, in general, is a locally compact and a-compact topological space with a countable base), and let ' denote a w-section of .Leg U D./be at most count- j=l able, measurable decomposition of .T hen U B 2(D) is a countable measur- j=l able decomposition of a Borel set B N(g).Consider a marked random measure with mark space g = [0,ee].The random measure is called a marked random measure modulated by the process Let Z i be a compound random measure with mark space g', Z i = , .X?)sr!i) obtained from the underlying counting measure N/= is,.!,i) by independent marking, and for each j let {X!I;i 1,2,...} be a sequence of independent and identi- cally distributed random variables with common mean cL/.Assume that N,i is a Pois- son counging measure directed by a random measure .A, i.e. (N, t) = (N, t) is a Cox process. he following is an extension of Lemma a.2 in Dshalalow [2].
Theorem 2. For the random measure Z modulated by the Cox process (N, M) and with the initial measure u, where E denotes the expected value with respect to probability measure P.
Let $ = [, Mli be a nonrandom absolutely continuous (with respect to the Le- besgue measure ) Borel measure, and 7j be its Radon-Nikodym density.Then, from Theorem 2 we have E[Ze] E i=o j I B/i(u)P{(u) e Di}(du) In particular, if for each j, the Poisson counting measure N/ is stationary, i.e. if 7j = A/(a positive constant), we have: E[Ze] = E o oA I B P(((u) e Dj}(du). (2) Definitions 3. Throughout the remainder of this paper we will consider the Borel r-algebra generaged by the usual topology on the real line.
(i) For every Borel set B and every Lebesgue-Stieltjes number [0 AB--,,(B) e ,co]   is called the total intensity of Z (over the set B with respect to ). measure 2., the (ii) If 2. is the restriction of the Lebesgue measure on (R+) and {Bt t } is a monotone increasing family of sets along a net g such that [.J tdBt-+, we call the number Definitions 4. (Dshalalow [2].) (i) Let Mt = i=oieTi be an irreducible and aperiodic Markov renewal process with a discrete state space for ,.Denote fl E*[T1] and fl-(ft, ;x e ).
Suppose that the embedded Markov chain is ergodic and that P-(p, ;x ) is its invariant probability measure.Mt is called recurrent-positive if its mean inter- renewal time, pflT, is finite.An irreducible and aperiodic and recurrent-positive Markov renewal process is called ergodic.
Theorem 5. Let Ml be an ergodic Markov renewal process and let Z a marked Poisson random measure modulated by the associated semi-Markov process .Denote fl=(/;xe), A-(A;x e ), a-(a;x e )) and p-a,fl,A, (the Hadamard product of vectors a, fl and )).Then, in a queueing system with the input Z , the following version of Little's formula holds true: = Proof.For some Borel set B denote P{(u) j}(du) O*(j B) = j" B O(B) = ((R)(j,B);j e q).Then, since is ergodic, for all x $, and it is equal to P*fl that pflT /t/rnoo e'([0, t]) exists; it is independent of x (For a reference, see Cinlar [].)This and formula (2) yield =lim E[Z([Ot])] ppT for the total intensity A of the input Z e modulated by the semi-Markov process LITTLE'S FORMULA FOR QUEUES WITH INPUTS MODULATED BY SEMI-REGENERATIVE PROCESSES Let Z e be a marked random measure modulated by an ergodic semi- regenerative process {ft, , (P*),v, (t); t >_ 0} (relative to the Markov renewal process {(,,T,}) with the stationary probability distribution r = (Try j ).Theorem 6.In a stochastic system with the input Z the following version of Little's formula for semi-regenerative modulated Poisson process holds true: The statement of the theorem is due the result in Dshalalow [o A lira z(B ) the total intensity of Z over +. 3. APPLICATIONS LITTLE'S FORMULA FOR QUEUES WITH INPUTS MODULATED BY SEMI-MARKOV PROCESSES.