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On the Superposition of m-Dimensional Point ProcessesAuthor(s): Erhan ÇinlarSource: Journal of Applied Probability, Vol. 5, No. 1 (Apr., 1968), pp. 169-176Published by: Applied Probability TrustStable URL: http://www.jstor.org/stable/3212084 .

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J. Appl. Prob. 5, 169-176 (1968) Printed in Israel

ON THE SUPERPOSITION OF m-DIMENSIONAL POINT PROCESSES

ERHAN CINLAR, Northwestern University, Evanston, Illinois

Summary Consider n independent vector valued point processes. Superposition is defined component by component as a natural extension of the definition for the one-dimensional case. Under proper conditions as n -+ oo, it is shown that the superposed process is a many-dimensional Poisson process with independent components. The results are applied to the superposition of Markov renewal processes.

1. Introduction A great majority of investigations in the theory of stores are based on the as-

sumption that the arrival process is a Poisson process. Moreover this assumption seems to be satisfied in real life more often than can be justified on theoretical grounds. Palm (1943) has made an attempt to explain facts of this kind by assum- ing that a given arrival process represents the superposition of a large number of independent stationary point processes of small intensity. Khintchine (1955) has completed the results of Palm, and proved that (under some broad assumptions) the resulting superposed process is very near a Poisson process. Goldman (1967) gives an elegant treatment of the problem and supplies necessary and sufficient conditions for the superposed process to approach a Poisson one. Ten Hoopen and Reuver (1966) have, considering the same problem, asserted the same result. (They show that the superposed stream has a constant renewal density function. Their assertion that this implies a Poisson process is, of course, not true.)

Recently, several authors (see Neuts (1966), Welch (1965), for example,) have studied queueing processes where the arriving customers are of different types so that the distribution of a service time depends on the type of customer being served. A usual assumption in these studies is that the arrivals of different types of customers form independent Poisson processes. Under these assumptions, the departures form a Markov renewal process whose state space is the Cartesian product of the set of customer types and the state space of the queue size process.

Received in revised form 4 September 1967. Supported in part by the National Science Foundation Grant GK-852.

169



170 ERHAN CINLAR

Since, in most cases, arrivals into a store are departures from other queueing sys- tems, the assumptions concerning the arrivals seem hard to justify.

In this paper we offer an explanation of facts of this kind by showing that if a

large number of independent vector valued point processes are superpcsed, under some broad assumptions, the resulting process will be a many-dimensicnal Poisson process. An application to the superposition of a large number Markov renewal process leads to results that are of particular interest in queueing theory.

In the next section a precise definition of the problem will be given. Section 3 contains the limit theorems characterizing the superposed process. In Section 4 the

special case of Markov renewal processes is given as an example.

2. The superposition problem Let I denote the set of non-negative integers. A right continuous non-decreasing

stochastic process {N(t),t ? 0} taking values in Im, where m is a fixed positive integer, is called an mn-dimensional point process on the real line. All prccesses considered in this paper are taken to be orderly: if t is a point of increase, then with probability one N(t) - N(t- ) = 1 for some j where 1j is the jth unit vector of I'".

We shall call a point process N stationary if the joint distribution of the vectors

N(t1) - N(to), -., N(t) - N(tk- 1) coincides with that of the vectors N(t, + a) - N(to + a),, N(tk --a) - N(tk- - +a) for any a >0, k 1 and 0 <to < t < ...<tk.

If N (N1, "", N,,) is an orderly and stationary point process, then each com-

ponent Nj as well as N = N + ... + N,,, is an orderly and stationary one-dimensional point process.

Consider a one-dimensional point process N with N(0) = 0. If N is staticnary, then

(1) v = lim t-'P{N(t) 1} t1 0

exists and is called the intensity of N, (cf. Khintchine (1955)). If, further, N is

orderly, then v is also the expected value of N(t + 1) - N(t). Stationarity of N

implies that, at an arbitrary epoch t, the time until the next jump of N has a distribution function which is independent of t. Further this distributicn function has the form

(2) P{N(x) > 1} = v [1 - B(y)] dy,

where B is a distribution function on (0, 0c) with mean 1 /v; (cf. Palm (1943), also Khintchine (1955). In some special cases, as for a stationary renewal process, B is the distribution function of the intervals between jumps. A final property for an orderly and stationary process will be needed (cf. Khintchine (1955))



On the superposition of m-dimensional point processes 171

(3) P{N(t) > 1} < vtB(t).

Consider n independent, orderly, and stationary rm-dimensional point processes. For the ith process N = (Ni1, Ni2,

"', Nim), define vi to be the intensity of N,j

and vi to be the intensity of Ni = N1 + .-.

+ Nim. (By the expected value in-

terpretation of the intensities, we have vi = vil + .'

+ Vim. Finally let Bi be the distribution function associated with Ni in the relation (2).

By the superposition of N1,' ".,Nn

we mean the mn-dimensional process L= (L1,. "",

Lm) where

Lj(t) = Nj(t) + +--- Nn(t)

for all j= 1,.., m and t> 0. Or, in vector notation, we simply write L= N1 +

*" + N,.

Let L = Li + ... + Lm; then we also have that L = N, + ... + N,. Now define

(4) 2j = vlj "' + Vnji, j = 1, "", m,

(5) 1 = vl + -lt

+ vn = A)L + '+2m.

The object of this paper is to show, under appropriate conditions, that as the number n of processes approaches infinity the superposed process L= (L1,

..-, L,) approaches an m-dimensional Poisson process. Throughout the dependence of L on the number n will be implicit.

We will start from the following assumptions:

(6) As n -+oo, sup vi -+0 while 2j and vij /vi remain constant for all i and j; 1 <i<n

(7) For any t>0, sup B(t)-+ 0 as n-+ oc. l <i<n

These assumptions have, roughly, the effect of transforming the time axis so that the intervals in any one process Ni become arbitrarily large. By requiring that

vij vi be kept constant we insure that this transformation of the time axis does not alter the relative status of a component Nj with respect to the other components Nik, k 0 j, of the same process Ni. For example in the case of stationary Markov renewal processes (see the last section), {vij/v;; j = 1, ..., m} is the stationary probability distribution of the underlying Markov chain for the ith process. Thus our assumption insures that the underlying Markov chains are not affected.

With the exception of our requirement regarding vij/vi, the conditions (6) and

(7) are both necessary and sufficient for L = LI + --- + Lm = N1 + --. + Nn

to

approach a Poisson process as n approaches infinity. This one-dimensional case is fully treated by Khintchine (1955). The recent treatment of this same problem by Goldman (1967) is for one-dimensional processes with k-dimensional Euclidean space as the parameter space. Our problem concerns m-dimensional processes with one-dimensional parameter space.



172 ERHAN CINLAR

Under the assumptions (6) and (7) it will be shown that as n -- co the processes

L1,L2,.".,Lm become independent Poisson processes with rates

,;,12,2' ,;,, respectively. Though requiring proof, the fact that Lj approaches a Poisson

process is not surprising. But, that L,,...,L,, become independent processes as n -+ oo is interesting in view of the fact that we do not assume independence of

Ni1, "",

Nim for any of the processes Ni.

3. The limit theorems

Throughout the following, the conditions (6) and (7) will be assumed to hold. We first list two lemmas: Lemma (8) is quite well known, the proof of Lemma

(9) (as well as that of Lemma (8)) can be found in Khintchine (1955), Chapter 5.

(8) Lemma. As n -+ co, P{L(t) = O} -+ e-'.

(9) Lemma. For any t > 0, 1•= P{N,(t) > 1} -+0 as n -+ cc.

For fixed non-negative integers k, ...,km

and t > 0, consider the events

(10) A = {L1(t) = k,'-",Lm(t)

= kIm)

(11) B = {Ni(t)< 1; i= 1,...,n}. For the fixed t, define

(12)- P{N,(t)= 1, N,(t) = 1 (12) j= P{N,(t) = 0}

Since the processes Ni are independent we can write

m kj ni (13) P(ArB)= r H JJfp jJ P{Nq(t)=O},

j=1 i=1 q=l

where Hf?f, fp, = 1 if kj = 0, and the summation is over the set of all com-

binations (pit'-,,Plk,;'";Pm1,'",P'mkm)

of kI + ""

+ km numbers taken from

{1,2, ...,n} .

(14) Theorem. For any t > 0, ks,-..,

km_ 2 0, as n -+ c0

P{L,(t) = k, ..,L,,(t)

= km} -+ e-t ('lt)ki (mt)k"'

Proof. Choose s > 0; using (2) and (3) and applying the condition (7) we

have, for sufficiently large n, that

(15) P{Ni(t) > 1} = vi J(1 - BS(x))dx

= vit - vtO1E,

(16) P(Ni(t) > 1} = vitO2e,,




where 01, 02 (and 03,04,"'"

to be used below) are fixed. Thus, P{Ni(t)= 1} = vt(1 - 03e). Since Bi,(t) < B,(t) always, (Bij corresponds to Nj in relation (2)), by a similar reasoning we have

(17) P{Nj(t) = 1} = vt( +

04e).

Using condition (6) regarding vij/vi remaining a constant, from (16),

(18) P{Nij(t) = 1, Ni(t) > 1} = vt02058 = ijt6;

and using the condition (6) regarding sup vi , from (15),

(19) P{Ni(t) = 0} = 1 - v,t + vtOle = 1 + 07e.

From (17) and (18) we can write

P{Ni,(t) = 1, N,(t) = 1} = P{N,(t) = 1} - P{Ni(t) = 1, N,(t) > 1} = vit( + 08e),

so that from (19) and definition (12) off,

(20) f j = vt(l + 096).

Hence, for sufficiently large n, m kj m kj

(21) n f,,j, = H II vpt ki+..+k,,(1 + 0e) j=1 i=1 j=1 i=1

where 0 is fixed. Now applying Lemma (25) to be proven below to (21),

m kj m

(22)

, - I- fpjH

tk+"+k

Hf j=1 i= 1 =1 kj '

Further, by Lemma (8),

(23) H-

P{Nq(t) = 0} = P{L(t) = 0} - e t' n -+ oO.

q=l

Combining (22) and (23), we get from (13) that

m (24) P(A n B) -+ e-A' H

(jit)kj/ki!, n -+ co. j=1

The theorem now follows by noting that, from the definition (11) of B and Lemma (9),

P(A) - P(A n B) = P(A n Bc) ? P(Bc) ? C P{N,(t) > 1} -~0, n - oo.

(25) Lemma. As n -+ co, • X= I- J= I-I j i 1j)k!



174 ERHAN CINLAR

Proof. Let Sm(k,-, ,km) be the indicated sum whcse limit as n - oc is our

object. For m = 1, k, 0 the Lemma follows from known results (cf. K hirtchire (1955)). Now suppose, for m > 1 and k,,

k,., kmi_ nonnegative integers, that

m- 1

(26) S,,inA(ki,...,k,,_1) = jH (Aj)k/k.i! + o(l), j I

and consider Sm(ki, ...km).

If k,,= 0, S,,(k, ", km)= Sm-_(k, .., k,, _ ) so

that the lemma holds. If km = 1, again the lemma is trivial in the light cf the induction hypothesis (26). Thus, suppose further that for km > 1,

(27) Sm(kl, ",

km-_, k1, - 1) H= - - o(1).

k! - (k, - 1) Now multiply each term of the sum Sm(k1, k,_- ki, - 1) by the sum of

all the vp

where p does not appear as a first subscript in that term; namely, mul-

tiply the term -

viT- k• ,,,•

by the quantity nm- 1 ki km- 1

(28) Q = Am- V ,

vp

.

j pim j=-1 i1 i=1

The result will be a sum of terms H-=i 1i 1 vi i . Each of these terms is one of the terms of the sum Sm(k, ,..., k); and further, any term of S,,(k1, ..., kin) ap- pears km times in this resulting sum.

If n is sufficiently large, for any combination of p's, we have

(29) _m_-

(k-- 1) Q

;m,

where k = k, + ... + kin. Thus, by the argument of the preceding paragraph,

(i2m - (k - 1)e)Sm(k

- - -, km_,, km - 1) ? kSm(k,,... k,i,) 1 ,,,iSm(ki,, ,km_-, 1k,,n -1).

Consequently from (27), since e is arbitrarily small for sufficiently large n,

S(,,,k) Sm(k, k).Sm(k, kmk- 1) -- (j '/k!, mn j=1

thus proving the lemma by induction. Theorem (14) above shows that, for any t > 0, the random variables

Ll(t), *.., L,,(t) of the superposed process in the limit are independent and Poisson

distributed with respective parameters ;, t, --., ,mt. Next we prove

(30) Theorem. As n --, o, the processes L ,,--,Lm

become independent Poisson

processes with intensities A ,"',2.m

respectively.

Proof. Let 0 = to < t, < t2 < < t,= t, (r 1), and define ui = i - ti-t , (i = l, . .-, r). Consider the event

A = (Lj(t,)- Lj(th-_)

= kjh; .j = 1,...,m, h = 1,-..,r}.




We need to show that

(31) P(A) -+ 1 I e-"J"(2juh)kJ/kjh!, n - o. j=1 h=l

Let B= {N,(t)< 1; i= 1 ...,n}, and set C = {Lj(t)= kj; j = 1,...,m} where

kj = i,=I

kjh, j = 1,...,m. Since the processes N,N2,2,.. are independent and

stationary,

P (A IB rn C ) =Q - kj, .

ul )kjU r

. .kj"'

+ o(

1 ) , m- k\I!... kj k\ tk t/

that is, conditional upon the occurrence of B and C, the probability of any one of the k, + ... + km points falling in the interval (ti, ti+ ] is ui/t independent of where the remaining points fall. Using Theorem (14) to obtain P(B C), in the limit as n -+ oo we have that

which gives (31).

4. Superposition of Markov renewal processes Consider a delayed Markov renewal process with state space

{1,2,.., m}, let

{Xn} be the underlying Markov chain and T1, T2, - the epochs of transitions. Let

Ny(t) be the number of pairs (Xn, T,) such that X, = j and T, ! t, n > 1.

Then, Nj(t) is the number of entrances to state j and N(t) = NI(t) + ... + Nm(t) is the total number of transitions in (0,t]. The m-dimensional process N =

(N,' ", Nm) is an orderly point process if T,+1 > T, with probability one

for all n. Each component of N is the counting process for a delayed renewal process. For more precise definitions, and proofs of the statement below, we refer to Pyke (1961), and Pyke and Shaufele (1966).

Let Qjk(t) = P{X.+I = k, T+1 - T tIX =j}, n 1; Pjk = Qk(O);

Rj(t)= kjk(t); m = tRj(dt) < co; and suppose j, rjPjk =

rk, i rk = 1 has

a unique solution. If

P{Xo = j} = 7rjmj/ i 7rm,,

and

P{Xi = k, T, < t Xo = j} = mnf [Pjk - Qjk(x)] dx,

then the m-dimensional point process N = (N, --., Nm) is a stationary one. The

intensities of Nj

and N = N1 + ... + Nm are, respectively, vj = tjr/ _irimi

and v =

1/-, 1rmi. The B function corresponding to N in (2) now is given by

B(t) = ~I r•niRi(t)

.



176 ERHAN CINLAR

Next consider n independent orderly and stationary Markov renewal processes

N,, --,N, and let L = N, + --- + N,, be their superposition. Under the conditions (6) and (7) our theorems imply that, as n - co, (L,,-..., L,) approaches an m-dimensional Poisson process. We note here that our requirement in (6) that vijvi be kept fixed is roughly equivalent to saying that the underlying Markov chain structure of the processes be left untouched.

For this special case where the processes being superposed are stationary Markov renewal processes, the conditions (6) and (7) can be replaced by the fol-

lowing single condition:

(32) As n -+ co, A`,, ', 4m remains constant while

sup R )(t) -+ 0, l <j<m

where R S is the distribution function of a stay in state j by the ith process (this, for the ith process, is the function Ri defined earlier).

Acknowledgement

I would like to thank the referee for his suggestions which have led to the present general form of the paper.

References

Cox, D. R. AND SMITH, W. L. (1954) On the superposition of renewal processes. Biometrika 41, 91-99.

GOLDMAN, J. R. (1967) Stochastic point processes: limit theorems. Ann. Math. Statist. 38, 771-779.

KHINTCHINE, A. I. (1955) Mathematical Methods in the Theory of' Queueing. Translation (1960), Griffin, London.

NEUTS, M. F. (1966) The single server queue with Poisson input and semi-Markov service times. J. Appl. Prob. 3, 202-230.

PALM, C. (1943) Intensitditsschwankungen im Fernsprechverkehr. Ericsson Techniks 44, 1-189.

PYKE, R. (1961) Markov renewal processes with finitely many states. Ann. Math. Statist. 32, 1243-1259.

PYKE, R. AND SHAUFELE, R. (1966) The existence and uniqueness of stationary measures for Markov renewal processes. Ann. Math. Statist. 37, 1439-1462.

TEN HOOPEN, M. AND REUVER, H. A. (1966) The superposition of random sequences of events. Biometrika 53, 383-389.

WELCH, P. D. (1965) On the busy period of a facility which serves customers of several types. J.R. Statist. Soc., B 27, 361-370.



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