Lajos Takács was born in Maglód, Hungary, on 21 August 1924. It was in this small town, 16 miles from Budapest, that he attended elementary school. He showed an early fascination with numbers, and worked on personal projects such as predicting the weight of the apricot harvest from 12 trees in the garden of the family home, by counting the buds on one tree in the spring.

A three-stage real diffusion process is used as a model of the T-cell count of an HIV-positive individual who is to receive antiviral therapy such as AZT. The ‘quality of life' of such a person is identified as the sojourn time of the diffusion process above a certain critical T-cell level c. The time of introducing therapy is defined as the first-passage time of the diffusion to a prescribed level z > c. The distribution of the sojourn time of the diffusion above the level c depends on the level z at which therapy is initiated. The expected sojourn time is explicitly computed as a function of z for the particular diffusion process defining the model. There is a simple criterion for determining when to start therapy as early as possible.

This paper first considers a simple deterministic epidemic in a growing population. It then examines a simple deterministic epidemic in several related growing communities, where infection is transmitted between individuals both within and across the communities. An explicit parametric solution is derived for the number of susceptibles in these communities as a function of time.

In more than four decades of prolific scientific activity, Lajos Takács has produced so much that no one contributor to this Festschrift can hope to cover all of it in a balanced way. I thus propose to make a virtue of necessity, and concentrate on those aspects of Takács' work which have particularly interested or influenced me, and on the impact Takács' ideas in these areas have had on the subsequent development of the subject.

We prove a generalization of Sanov's theorem in which the state space S is arbitrary and the set of probability measures on S is endowed with the τ -topology.

Let M be a random matrix chosen from Haar measure on the unitary group Un. Let Z = X + iY be a standard complex normal random variable with X and Y independent, mean 0 and variance ½ normal variables. We show that for j = 1, 2, …, Tr(Mj) are independent and distributed as √jZ asymptotically as n →∞. This result is used to study the set of eigenvalues of M. Similar results are given for the orthogonal and symplectic and symmetric groups.

Let A1, A2, …, An and B1, B2,. ., BN be two sequences of events on the same probability space. Let mn(A) and mN(B), respectively, be the number of those Aj and Bj which occur. Let Si,j denote the joint ith binomial moment of mn(A) and jth binomial moment of mN(B), 0 ≤ i ≤ n, 0 ≤ j ≤ N. For fixed non-negative integers a and b, we establish both lower and upper bounds on the distribution P(mn(A) = r, mN(B) = u) by linear combinations of Si,j, 0 ≤ i ≤ a, 0 ≤ j ≤ b. When both a and b are even, all mentioned S¡,j are utilized in both the upper and the lower bound. In a set of remarks the results are analyzed and their relation to the existing literature, including the univariate case, is discussed.

Statistical exponential families invariant with respect to the groups of the cone or the paraboloid of revolution are discussed. B(x, y) denotes the symmetric bilinear form on x0y0 – x1y1 – ·· ·– xdyd on ℝd+1, C denotes the cone of revolution in ℝd+1 {x; B(x,x) > 0 and x0 > 0}, and, for p > ½(d−1), μ p is the positive measure on ℝd+1 defined by its Laplace transform ⨍ exp (B(x,y))μp(dy) = (B(y,y))−p for y on C. More precisely, if p > ½ (d−1) one has and μ½(d−1) concentrated on the boundary ∂C. This paper studies the natural exponential families

Hierarchical information networks are important in applications where the information management must support an existing tree-structured organization. Embedded computer-communication systems in military applications, with a dominant hierarchical command structure, are the most prominent examples. Also typical of such applications is the variability in message demand (both sources and intensity) depending on the external conditions encountered by the encapsulating system (the system supported by the embedded computer-communication system). Using a queueing network model for a hierarchical information network, we compare the effect of limited dynamic reconfiguration on expected transmission delays. The limited reconfigurability takes the form of apex transition among a proper subset of the communication nodes designated as the apex candidate set. Each apex candidate can assume the ultimate position under designated conditions. This network architecture is called a dynamic hierarchy. The model includes N + 1 nodes (0, …, N) with 0 identifying the apex node. We assume that message processing at each node is described by an M/M/1 model (single server with Poisson arrivals and exponential service times). Further message transfers among the nodes are served by communication links which also behave as M/M/1 queues.

Two distinctive features characterize the queueing network model:

1. The assignment of a set of weights to the nodes dependent on the hierarchical level reflects the increasing importance of information as it is transferred to higher levels.

2. The dynamic hierarchy requires a communications protocol that partitions the analysis of network delay into three periods: regular operation, reconfiguration, and adjustment. Characterization of the performance of the dynamic hierarchy entails the description of message transmission delay as a composite of the three periods.

In this contribution we consider an M/M/1 queueing model with general server vacations. Transient and steady state analysis are carried out in discrete time by combinatorial methods. Using weak convergence of discrete-parameter Markov chains we also obtain formulas for the corresponding continuous-time queueing model. As a special case we discuss briefly a queueing system with a T-policy operating.

We consider the standard single-server queue with unlimited waiting space and the first-in first-out service discipline, but without any explicit independence conditions on the interarrival and service times. We find conditions for the steady-state waiting-time distribution to have asymptotics of the form x–1 log P(W> x) → –θ ∗as x → ∞for θ ∗ > 0. We require only stationarity of the basic sequence of service times minus interarrival times and a Gärtner–Ellis condition for the cumulant generating function of the associated partial sums, i.e. n–1 log E exp (θSn) → ψ (θ) as n → ∞, plus regularity conditions on the decay rate function ψ. The asymptotic decay rate θ is the root of the equation ψ (θ) = 0. This result in turn implies a corresponding asymptotic result for the steady-state workload in a queue with general non-decreasing input. This asymptotic result covers the case of multiple independent sources, so that it provides additional theoretical support for a concept of effective bandwidths for admission control in multiclass queues based on asymptotic decay rates.

For a network of G/∞ service facilities, the transient joint distribution of the facility populations is shown by new simple methods to have a simple Poisson product form with simple explicit formulas for the means. In the network it is assumed that: (a) each facility has an infinite number of servers; (b) the service time distributions are general; (c) external traffic is non-homogeneous in time; (d) arrivals have random or deterministic routes through the network possibly returning to the same facility more than once; (e) arrivals use the facilities on their route sequentially or in parallel (as in the case of a circuitswitched telecommunication network). The results have relevance to communication networks and manufacturing systems.

We consider single-server queueing systems that are modulated by a discrete-time Markov chain on a countable state space. The underlying stochastic process is a Markov random walk (MRW) whose increments can be expressed as differences between service times and interarrival times. We derive the joint distributions of the waiting and idle times in the presence of the modulating Markov chain. Our approach is based on properties of the ladder sets associated with this MRW and its time-reversed counterpart. The special case of a Markov-modulated M/M/1 queueing system is then analysed and results analogous to the classical case are obtained.

Lagrangian distributions are reviewed from the viewpoint of the Galton-Watson process. They are related to the busy period in queuing systems and to the first visit in random walks.

A property of the distributions is remarked for the application to vacant vehicles in a new transit system. Combinatorial identities of multinomial and binomial coefficients and related recurrences are shown by a probabilistic method. Based on the identities and recurrences, random forests generated by the Poisson and geometric Galton-Watson processes are characterized.

Let X, X1, X2, … be i.i.d. Sn=Σ1nXj, E|X| > 0, E(X) = 0 and τ = inf {n ≥ 1: Sn ≥ 0}. By Wald's equation, E(τ) =∞. If E(X2) <∞, then by a theorem of Burkholder and Gundy (1970), E(τ1/2) =∞. In this paper, we prove that if E((X–)2) <∞, then E(τ1/2) =∞. When X is integer-valued and X ≥ −1 a.s., a necessary and sufficient condition for E(τ1–1/p) <∞, p > 1, is Σn–1–1p E|Sn| <∞.

For positive recurrent nearest-neighbour, semi-homogeneous random walks on the lattice {0, 1, 2, …} X {0, 1, 2, …} the bivariate generating function of the stationary distribution is analysed for the case where one-step transitions to the north, north-east and east at interior points of the state space all have zero probability. It is shown that this generating function can be represented by meromorphic functions. The construction of this representation is exposed for a variety of one-step transition vectors at the boundary points of the state space.

Some exact and asymptotic joint distributions are given for certain random variables defined on the excursions of a simple symmetric random walk. We derive appropriate recursion formulas and apply them to get certain expressions for the joint generating or characteristic functions of the random variables.

Certain last-exit and first-passage probabilities for random walks are approximated via a heuristic method suggested by a ladder variable argument. They yield satisfactory approximations of the first- and second-order moments of the queue length within a busy period of an M/D/1 queue. The approximation is applied to the wider class of random walks that arise in studying M/GI/1 queues. For gamma-distributed service times the queue length distribution is independent of the arrival rate. For other distributions where the arrival rate affects the queue length distribution, we have to use conjugate distributions in order to exploit a local central limit property. The limit underlying the approximation has the nature of a Brownian excursion.

The source of the problem lies in recent queueing inference work; the connection with Takács' interests comes from both queueing theory and the ballot theorem.

This article investigates the accuracy of approximations for the distribution of ordered m-spacings for i.i.d. uniform observations in the interval (0, 1). Several Poisson approximations and a compound Poisson approximation are studied. The result of a simulation study is included to assess the accuracy of these approximations. A numerical procedure for evaluating the moments of the ordered m-spacings is developed and evaluated for the most accurate approximation.