JOINT SIMULATION OF BACKWARD AND FORWARD RECURRENCE TIMES IN A SUPERPOSITION OF INDEPENDENT RENEWAL PROCESSES C.Y. Teresa Lam Department of Industrial & Operations Engineering University of Michigan Ann Arbor, MI 48109-2117 Technical Report 90-11 April 1990 Revised August 1990

Joint Simulation of Backward and Forward Recurrence Times in a Superposition of Independent Renewal Processes C. Y. Teresa Lam Department of Industrial and Operations Engineering The University of Michigan Ann Arbor, MI 48109 Abstract It is shown that, in a superposition of finitely many independent renewal processes, an observation from the limiting (when t -+ oo) joint distribution of backward and forward recurrence times at t can be simulated by simulating an observation of the pair (UW, (1-U)W), where U and W are independent random variables with U -uniform(0,1) and W distributed according to the limiting total life distribution of the superposition process. SUPERPOSITION OF RENEWAL PROCESSES; LIMITING TOTAL LIFE DISTRIBUTION; SIMULATION 1 Introduction and Summary Winter (1989) showed that in a renewal process with interarrival distribution F, an observation from the limiting (when t -- oo) joint distribution of backward and forward recurrence times at t can be simulated by simulating an observation of the pair (UW, (1 - U)W), where U and W are independent random variables with U -uniform(0,1) and W distributed according to the length-biased version of F or the limiting total life distribution. It is shown in this note that the analogous result holds for the superposition of independent renewal processes. 1

Suppose that there are p independent ordinary renewal processes in operation simultaneously. Let Ft, i = 1,2,...,p, be the probability distribution functions for the successive interarrival times of the ith process with Fi(O) = 0 and positive finite mean ti. Furthermore, let us assume that Fi is absolutely continuous for all i = 1,2,.... The above conditions ensure that for all _component processes, with probability one a non-zero time is spent between transitions. Also, simultaneous occurrence of events in the superposed process has probability zero. Consider the sequence of events formed by pooling the individual processes. In general, the superposed process is not a renewal process. At the time when an event occurs in the superposition process, one of the processes, say process i, probabilistically starts over. In addition, the others have age D,..., D.i-1, D+1,..., D respectively, where the Dj s are random variables. The age here refers to the time since the last event occurs in a particular component process. Suppose that at time 0, the component processes all have age zero. Let yt and 1t be the forward recurrence time and the total life at time t of the superposed process respectively. From Lam and Lehoczky (1991), for i = 1,...,p, if Fi satisfies all the conditions stated above and furthermore, it is non-arithmetic, then the limiting distributions of the forward recurrence time and the total life for the superposed process exist and are given by,:~f I[1/(z ~(1 -Fi(x))dx] ifz>O limP (Yt>Z) = Z{ L iZ ) ())] ifz0 (1) t-ooI [t~~ ~1 ~otherwise and 1 x dG(x) z>0 lim P(Pt t z) = K(z)= (Oz] (2) t 0 otherwise where = (i)/(i) - Fi(x-)) j [I (-Fj(Y if > 0 G(x) - W: j:~,3,) J (3) 0 otherwise and = [ -. From Lam and Lehoczky (1991), Results (1) and (2) above also hold in the superposition of p independent delayed renewal processes. Furthermore, if 6t is the backward recurrence time at time t of the superposed process, then 2

for y, z > O z z lim P(t > y and t > z) = lim P(t-y > Y + z) = (1 - F(x))dx). t-o t-oo - i J(y+z,-o) (4) We are now ready to state and prove the following result which is the analogue of the result given in Winter (1989). Theorem 1 Let U and W be two independent positive random variables, with U.uniform(O, 1) and -W f K with K as in Result (2). Then P(UW > y, (1 - U)W > z) = i=i:/(y+z,) 0 otherwise (5) Proof: Let X1,...,Xp be independent random variables such that Xi, i = 1,...,p, follows that limiting forward recurrence or backward recurrence distribution of the ith component process. Also, let T = min{Xl,...,Xp}. As in Winter (1989), for y,z > 0, P(UW > y, (1- U)W > z) = p(W > y + Z < U < 1- ) W W = / (1- G(u)) du lJy+z,ox>) (6) By substituting Equation (3) have for y,z > 0, into (6) and using the definitions of X, i = 1,...p and T, we P(UW >y,(1-U)W > )= I(y+zo) (1- i(u )) +z,oo ) i= Pi j=l, ( - Fj(w)) dwdu p = ZP(y+z<Xi<minXj) i=1 j = P(T>y+z)=iJ - (1- F(w))dw i=1 -i (y+z,00) (7) This completes the proof of Theorem 1. 3

2 Concluding Remarks Theorem 1 and Equation (4) above show how one can simulate a joint observation of the pair (6t, t) for very large t. It can be obtained by computing (uw, (1 - u)w), where u is a simulated observation of U ~uniform(0,1) and w is a simulated observation of W I K, with U and W independent. This is, of course, practical only when the Fi s are such that the distribution function K is easily derived. This includes the cases when Fi = F for all i = 1,2,..., p and F is uniform, shifted exponential, gamma or beta. In particular, the results below are easily verified. 1. When Fi = F, uniform(0, 1), 0 if x < 0 G(x) = 1- (1 - x)2p- if0 < <1 (8) 1 x>l and 0 if z < O K(z) = < 1-(1-z)2P-l[ +(2p- i)z] if0 < z < 1 (9) 1 if z > 1 2. When Fi= F, 1 -exp[-A(x - 1)] if > 1 (10) 0 otherwise and A > O. We have after some tedious but straightforward calculations, 0 if x <0 G(x) = 1- +A [1 + A(1 - )]P-1 if < (11) 1-( +)1 ) exp[-Ap(x - 1)] if x> 1 4

(o if z < O 1q n+ (-z) P- 1+ Ap+( ) i+ O<< AK () = +I (12) 1 P):1 + ApI 1- + 1+ 1 + AIA +(1 + A)- [1 +P -(1 + Ap+ Ap(z - 1))exp(-Ap(z - 1))] if z > 1 Similarly, when F is gamma or beta, we can derive both distribution functions G and K. The calculations may be tedious but they involve only integrations of polynomials on bounded intervals when F is beta. In the case when F is gamma, we require to evaluate definite integrals of product of exponential functions and polynomials. These integrals can be calculated easily using integration by parts. Acknowledgement I would like to thank the referee whose valuable comments helped improve the presentation of results in this paper. References [1] Lam, C. Y. T. and Lehoczky J. P. (1991) Superposition of renewal processes. To appear in Advances in Applied Probability. [2] Winter, B. B. (1989) Joint Simulation of Backward and Forward Recurrence Times in a Renewal Process. Journal of Applied Probability, Vol. 26, No. 2, 404-407. 5