In a queuing process, let 1/λ be the mean time between the arrivals of two consecutive units, L be the mean number of units in the system, and W be the mean time spent by a unit in the system. It is shown that, if the three means are finite and the corresponding stochastic processes strictly stationary, and, if the arrival process is metrically transitive with nonzero mean, then L = λW.

ABSTRACT

ABSTRACT

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A "branch and bound" algorithm is presented for solving the traveling salesman problem. The set of all tours (feasible solutions) is broken up into increasingly small subsets by a procedure called branching.For each subset a lower bound on the length of the tours therein is calculated. Eventually, a subset is found which contains a single tour whose length is less than or equal to some lower bound for every tour.The motivation of the branching and the calculation of the lower bounds are based on ideas frequently used in solving assignment problems.Computationally, the algorithm extends the size of problem which can reasonably be solved (i.e. the size of problem for which a guaranteed optimal tour can be found without using methods special to the particular problem). The algorithm has been programmed for an IBM 7090.Problems constructed by using 3 digit random numbers as distances between cities have required average computing times as follows: 10