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.
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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
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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.
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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.
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