In this paper we focus on networks of infinite-server queues with nonhomogeneous Poisson arrival processes. We start by introducing a more general Poisson-arrival-location model (PALM) in which arrivals move independently through a general state space according to a location stochastic process after arriving according to a nonhomogeneous Poisson process. The usual open network of infinite-server queues, which is also known as a linear population process or a linear stochastic compartmental model, arises in the special case of a finite state space. The mathematical foundation is a Poisson-random-measure representation, which can be obtained by stochastic integration. It implies a time-dependent product-form result: For appropriate initial conditions, the queue lengths (numbers of customers in disjoint subsets of the state space) at any time are independent Poisson random variables. Even though there is no dependence among the queue lengths at each time, there is important dependence among the queue lengths at different times. We show that the joint distribution is multivariate Poisson, and calculate the covariances. A unified framework for constructing stochastic processes of interest is provided by stochastically integrating various functionals of the location process with respect to the Poisson arrival process. We use this approach to study the flows in the queueing network; e.g., we show that the aggregate arrival and departure processes at a given queue (to and from other queues as well as outside the network) are generalized Poisson processes (without necessarily having a rate or unit jumps) if and only if no customer can visit that queue more than once. We also characterize the aggregate arrival and departure processes when customers can visit the queues more frequently. In addition to obtaining structural results, we use the stochastic integrals to obtain explicit expressions for time-dependent means and covariances. We do this in two ways. First, we decompose the entire network into a superposition of independent networks with fixed deterministic routes. Second, we make Markov assumptions, initially for the evolution of the routes and finally for the entire location process. For Markov routing among the queues, the aggregate arrival rates are obtained as the solution to a system of input equations, which have a unique solution under appropriate qualifications, but not in general. Linear ordinary differential equations characterize the time-dependent means and covariances in the totally Markovian case.

Networks of infinite-server queues with nonstationary Poisson input

We consider Markov models for growth of populations subject to catastrophes. Emphasis is placed on discrete-state models where immigration is possible and the catastrophe rate is population-dependent. Explicit formulas for descriptive quantities of interest are derived when catastrophes reduce population size by a random amount which is either geometrically, binomially or uniformly distributed. Comparison is made with continuous-state Markov models in the literature in which population size evolves continuously and deterministically upwards between random jumps downward.

/pdf/birth-immigration-and-catastrophe-processes-44478ty74l.pdf

Birth, immigration and catastrophe processes

A revealing alternate proof is provided for the Iglehart (1965), (1973)–Borovkov (1967) heavy-traffic limit theorem for GI/G/s queues. This kind of heavy traffic is obtained by considering a sequence of GI/G/s systems with the numbers of servers and the arrival rates going to ∞ while the service-time distributions are held fixed. The theorem establishes convergence to a Gaussian process, which in general is not Markov, for an appropriate normalization of the sequence of stochastic processes representing the number of customers in service at arbitrary times. The key idea in the new proof is to consider service-time distributions that are randomly stopped sums of exponential phases, and then work with the discrete-time vector-valued Markov chain representing the number of customers in each phase of service at arrival epochs. It is then easy to show that this sequence of Markov chains converges to a multivariate O–U (Ornstein–Uhlenbeck) diffusion process by applying simple criteria in Stroock and Varadhan (1979). The Iglehart–Borovkov limit for these special service-time distributions is the sum of the components of this multivariate O–U process. Heavy-traffic convergence is also established for the steady-state distributions of GI/M/s queues under the same conditions by exploiting stochastic-order properties.

On the heavy-traffic limit theorem for GI/G/∞ queues

Kendall's (1956) approach to the ‘general’ epidemic is generalized by dropping the assumptions of constant infectivity and random recovery or death of ill individuals. A great deal of attention is paid to the biological background and the heuristics of the model formulation. Some new results are: (l) the derivation of Kermack's and McKendrick's integral equation from what seems to be the most general set of assumptions in section 2.2, (2) the use of Kermack's and McKendrick's final value equation to arrive at a finite time version of the threshold theorem for the general case, comparable to that for the case of only one Markovian state of illness in section 2.5, (3) the analysis of the behaviour of the solutions of the integral equation when the starting infection approaches zero in section 2.7, (4) the derivation of the probability structure of a general branching process, after conditioning on extinction in section 3.6, (5) the statement of the generalized versions of Kendall's ideas in the form of precise limit conjectures in section 4, (6) the derivation of a closed expression for the limit epidemic resulting from (3) in appendix 4.

The epidemic in a closed population with all susceptibles equally vulnerable; some results for large susceptible populations and small initial infections.

The distribution of the extinction time for a linear birth and death process subject to catastrophes is determined. The catastrophes occur at a rate proportional to the population size and their magnitudes are random variables having an arbitrary distribution with generating function d(·). The asymptotic behaviour (for large initial population size) of the expected time to extinction is found under the assumption that d(.) has radius of convergence greater than 1. Corresponding results are derived for a related class of diffusion processes interrupted by catastrophes with sizes having an arbitrary distribution function.

The extinction time of a birth, death and catastrophe process and of a related diffusion model

A population process is considered where particles reproduce according to an age-dependent branching process, and are subjected to disasters which occur at the epochs of an independent renewal process. Each particle alive at the time of a disaster, survives it with probability p and the survival of any particle is assumed independent of the survival of any other particle. The asymptotic behavior of the mean of the process is determined and as a consequence, necessary and sufficient conditions are given for extinction.

A branching process with disasters

Some necessary and sufficient conditions are found for the existence of a proper non-degenerate limiting distribution for a Bellman-Harris age-dependent branching process with a compound renewal immigration component. A number of these results are applicable to the batch arrival GI / G / ∞ queueing process. Some aspects of the situation when there is no such limiting distribution are considered. The situation when the immigration component is a nonhomogeneous compound Poisson process is briefly considered.

On the subcritical Bellman-Harris process with immigration

A well-known theorem in the theory of branching random walks is shown to hold when only Σ j log jp j

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/7AAD01982A3B63996965045459DA2C15/S0021900200022907a.pdf/div-class-title-a-note-on-the-branching-random-walk-div.pdf

A note on the branching random walk

The results of this paper deal directly with the behavior of compositions of random probability generating functions. These results are then applied to the theory of branching processes with random environments. In particular, it is shown that it is not possible to strengthen many of the existing limit theorems on this subject. BRANCHING PROCESS; BRANCHING PROCESS WITH RANDOM ENVIRONMENT; RANDOM ENVIRONMENT; RANDOM PROBABILITY GENERATING FUNCTIONS

A theorem on compositions of random probability generating functions and applications to branching processes with random environments

Consider a finite urn scheme with balls uniformly distributed amongst the urns. Using a technique of Karlin, the asymptotic behavior of the number of throws necessary to obtain at least m (m -? 1) balls in a given fraction of the urns is studied. FINITE URN SCHEME; POISSON APPROXIMATION Consider the following problem. Balls are thrown into n urns uniformly and independently. Let Nm(n) equal the number of throws necessary to,obtain at least m balls in each urn. Erd6s and R6nyi [1] proved that for any 0 O, j > 0, qj (z) = e--zi/y!. Received in revised form 4 August 1976. Research supported in part by the Institute of Mathematical Statistics, University of Copenhagen and by National Science Foundation Grant MPS 74-18967. 212 This content downloaded from 157.55.39.104 on Mon, 20 Jun 2016 06:10:39 UTC All use subject to http://about.jstor.org/terms A generalization of a result of Erdis and Renyi 213 We first give a heuristic proof of Theorem 1. Define: X,(i) = numbers of balls in urn j after i throws, 15j-n, i>1. (y) = if y 0, Y(O) denotes a Poisson variable with parameter 0. Let X, and X2 be any two non-negative integer-valued random variables, and define d(X, X2) = sup P(X, E A)P(X2E B)J, where A ranges over all subsets of the non-negative integers. For a discussion of the metric d, the reader is referred to [2]. The two properties of d that we will need are: (4) d (Xi, X2): P(X, 7 X2), and if X1 and X2 are Poisson with parameters 01 and 02, then (5) d (Xl, X2) ~0, 02 * Proofs of (4) and (5) are given in [2]. We then have the following proposition. Proposition 1 (Hodges and Le Cam [5]). Let B1, . ", B, be n independent events and let N = If_ IB, be the number which occurs. Let This content downloaded from 157.55.39.104 on Mon, 20 Jun 2016 06:10:39 UTC All use subject to http://about.jstor.org/terms

Norman Kaplan

Papers

A branching process with disasters

On the subcritical Bellman-Harris process with immigration

A note on the branching random walk

A theorem on compositions of random probability generating functions and applications to branching processes with random environments

A generalization of a result of Erdös and Rényi