A model for interaction of a Poisson and a renewal process and its relation with queuing theory

This paper discusses the response process when a Poisson process interacts with a renewal process in such a way that one or more points of the Poisson process eliminate a random number of consecutive points of the renewal process. A queuing situation is devised such that the c.d.f. of the length of the busy period is the same as the c.d.f. of the length of time intervals of the renewal response process. The Laplace-Stieltjes transform is obtained and from this the expectation of the time intervals of the response process is derived. For a special case necessary and sufficient conditions for the response process to be a Poisson process are found. RENEWAL PROCESS; POISSON PROCESS; INTERACTION; QUEUING THEORY; FINITE WAITING ROOM 1. The model and its connection with queuing theory Ten Hoopen and Reuver [2] have suggested the following kind of interaction between two random point processes as a mathematical model for neuron firing. Points in a renewal process, the excitatory process, are eliminated by points in a Poisson process, the inhibitory process, according to the following rule: whenever one or more points of the inhibitory process arrive, the next point of the excitatory process is eliminated. In this paper this model is generalized by assuming that one or more points of the inhibitory process eliminate a random number of points of the excitatory process. Ten Hoopen and Reuver's model has been generalized by Coleman and Gastwirth [1] in a different way: the points of the inhibitory process are effective during a time period of constant or random length, during which they eliminate all arriving points of the excitatory process. We assume in the following that the excitatory process is a renewal process and denote by F the c.d.f. of the time intervals between points in this process. Furthermore, we assume that the inhibitory process is a Poisson process with intensity A and that one or more points of the inhibitory process eliminate the next k points of the excitatory process with probability Pk, 0 Pk = 1. We also assume that points of the inhibitory process arriving during a time period when points are eliminated by previous points of the inhibitory process, do not affect the elimination process. The case studied by Ten Hoopen and Reuver is characterized by pl = 1. Received in revised form 28 July 1971. 451 This content downloaded from 157.55.39.17 on Wed, 31 Aug 2016 04:41:49 UTC All use subject to http://about.jstor.org/terms 452 LENNART RADE Under these assumptions the random process of remaining points of the excitatory process, the response process, is a renewal process. We denote by F, the c.d.f. of the time intervals between points of this process. There is a close connection between the problem of determining the c.d.f. F, of the response process for a given c.d.f. F of the renewal excitatory process and a given elimination rule and queuing theory. As a matter of fact, the first problem can be interpreted as a problem of determining the busy period of a suitably chosen queuing situation. Consider for instance the case originally studied by Ten Hoopen and Reuver, when one or more points of the inhibitory process eliminate exactly one point of the excitatory process. In this case F, is the c.d.f. of the length of the busy period for the M/G/1 queuing system, where the arrival process is a Poisson process with intensity ), the service times have c.d.f. F and there is a finite waiting room with space for one waiting customer. For the case studied in [1] by Coleman and Gastwirth, when the inhibitors are effective during a random time period, one has to consider a similar queuing situation but with the waiting time of the customers limited by a random variable. For the case studied in this paper the appropriate queuing situation is an M/G/1 system with the following characteristics. The arrival process is a Poisson process with intensity A^ and there is room for only one waiting customer. The busy period is initiated by a virtual (imaginary) customer whose service time has c.d.f. F, while the subsequent customers have a service time with c.d.f. ,km=0 PkF*, where F[ is the convolution of k functions F; the latter service time can be considered as consisting of k phases with probability Pk. Arrivals are permitted only during the service time of the virtual customer, and during the last phase of the service of the actual customer, but at no other time. Put P,(x) = Probability that the length of the busy period is at most x, and the number of customers (including the virtual customer) served during the busy period is n. We now get PA(x) = fe-tdF(t) + Po (1 e-t) dF(t), P(x) = (1 e-'t')P,,1(xtl t2) dF(tl) d Pk_ l(t2) , n> 1. 1t=0 2=0 k=1 Taking Laplace-Stieltjes transforms yields (we denote by f the Laplace-Stieltjes transform off) P1(s) = P(s + o) + po(i(s)[(s + 2)), 00 P(s) = P~1(s)((s)P(s + )) pk p,(s)"k, ,1 > I k=l This content downloaded from 157.55.39.17 on Wed, 31 Aug 2016 04:41:49 UTC All use subject to http://about.jstor.org/terms A modelfor interaction of a Poisson and a renewalprocess 453 But Fi(x) = ~,P= 1P(x) and thus Pl(s) = ~~, 1P(s). Summation of the formulas above then gives P(s) = Poo(s) + (1 po)(s + ) PA(s) (+(s) f(s +)) pkF(S)k-1 k=