The asymptotic joint distribution of the state occupancy times in an alternating renewal process

An alternating renewal process starts at time zero and visits states 1,2,…,r, 1,2, …,r 1,2, …,r, … in sucession. The time spent in state i during any cycle has cumulative distribution function F_i, and the sojourn times in each state are mutually independent, positive and nondegenerate random variables. In the fixed time interval [0,T], let U_i(T) denote the total amount of time spent in state i. In this note, a central limit theorem is proved for the random vector (U_i(T), 1 ≤ i ≤ r) (properly normed and centered) as T → ∞.     

Book reviews

Managing International Conflict, From Theory to Practice: a Teaching Tool Using CASCON by Lincoln P. Bloomfield and Allen Moulton, New York: St Martin's Press, 1997, ISBN 0-312-13675-7 (pbk), £18.99.

The American Century: the Rise and Decline of the United States as a World Power by Donald W. White, London: Yale University Press, 1997, ISBN 0-300-05721-0 (hbk), £25.

Humanitarian Intervention in Contemporary Conflict by Oliver Ramsbotham and Tom Woodhouse, Cambridge: Blackwell, 1996.

The US Military/NGO Relationship in Humanitarian Interventions by Chris Seiple, Peacekeeping Institute, Center for Strategic Leadership, US Army War College, 1996.

Uncivil Wars: International Security and the New Internal Conflicts by Donald M. Snow, London: Lynne Rienner Publishers, 1996, ISBN 1-55587-655-2 (pbk), £15.50, ISBN 1-55587-648-X (hbk), £33.50.

Intelligence Power in Peace and War by Michael Herman, Royal Institute for International Affairs, Cambridge: Cambridge University Press, 1996, ISBN 0-521-56231-7 (hbk), £50, ISBN 0-521-56636 (pbk), £16.95.

UK Eyes Alpha: Inside British Intelligence by Mark Urban, London: Faber and Faber, 1996, ISBN 0-571-17689-5 (hbk), £16.99.

New Cloak, Old Dagger by Michael Smith, London: Victor Gollancz, 1996, ISBN 0-575-06150-2 (hbk), £20.

Conversion Survey 1997: Global Disarmament and Disposal of Surplus Weapons by Bonn International Center for Conversion, Oxford: Oxford University Press, 1997.

Plutonium and Highly Enriched Uranium 1996: World Inventories, Capabilities and Policies by David Albright, Frans Berkhout and William Walker, Oxford: SIPRI and Oxford University Press, 1997.

The Wars of Eduard Shevardnadze by Carolyn McGiffert Ekedahl and Melvin A. Goodman, University Park, PA: Penn State University Press, 1997.     

Extrapolation of the mean lifetime of a large population from its preliminary survival history

A large population of independent identical units having finite mean lifetime T is observed. From the history A(y) of cumulative arrivals and the history B(y) of cumulative removals in the interval 0 ≦ y ≦ τ one must predict at time τ the desired T . Two lifetime predictors X(τ) and Y(τ) and related simple predictors obtained from A(y) and B(y) are shown to converge to T with a rate of convergence dependent on the structure of the failure rate function of the units. This dependence is studied theoretically and numerically.     

A method for reducing coefficients in integer linear inequalities

Approaches are considered for the reduction of coefficients in linear integer inequalities. It is shown that coefficients may be reduced easily in many practical examples. The (0-1) problem is also reconsidered and certain areas of exploration developed.     

Some limiting distributions associated with sequences of multinomial trials

A series of independent trials is considered in which one of k ≥ 2 mutually exclusive and exhaustive outcomes occurs at each trial. The series terminates when m outcomes of any one type have occurred. The limiting distribution (as m → ∞) of the number of trials performed until termination is found with particular attention to the situation where a Dirichlet distribution is assigned to the k vector of probabilities for each outcome. Applications to series of races involving k runners and to spares problems in reliability modeling are discussed. The problem of selecting a stopping rule so that the probability of the series terminating on outcome i is k^?1 (i.e., a “fair” competition) is also studied. Two generalizations of the original asymptotic problem are addressed.     

A game theoretic approach to a two firm bidding problem

In this paper a model is developed for determining optimal strategies for two competing firms which are about to submit sealed tender bids on K contracts. A contract calls for the winning firm to supply a specific amount of a commodity at the bid price. By the same token, the production of that commodity involves various amounts of N different resources which each firm possesses in limited quantities. It is assumed that the same two firms bid on each contract and that each wants to determine a bidding strategy which will maximize its profits subject to the constraint that the firm must be able to produce the amount of products required to meet the contracts it wins. This bidding model is formulated as a sequence of bimatrix games coupled together by N resource constraints. Since the firms' strategy spaces are intertwined, the usual quadratic programming methods cannot be used to determine equilibrium strategies. In lieu of this a number of theorems are given which partially characterize such strategies. For the single resource problem techniques are developed for determining equilibrium strategies. In the multiple resource problem similar methods yield subequilibrium strategies or strategies that are equilibrium from at least one firm's point of view.