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In this paper we get some sufficient conditions for the finiteness or nonfiniteness of the passage-time moments for nonnegative discrete parameter processes. The developed criteria are closely connected with the well-known results of Foster for the ergodicity of Markov chains and are Ž . given in terms of sub super martingales. Then, as an application of the obtained results, we get explicit conditions for the finiteness or nonfiniteness of passage-time moments for reflected random walks in a quadrant with zero drift in the interior.

A single-server processor-sharing system with M job classes is analyzed in the steady state. The scheduling strategy considered divides the total processor capacity in unequal fractions among the different job classes. More precisely, if there are N~jobs of classj in the system, j = 1, 2 ..... M, each class k job receives a fraction gh/(~M.~ giN~) of the processor capacity.Earlier analyses of this system are shown to be incorrect and new expressions for the conditional expected response times Wk(t) of class k jobs with required service time t are obtained (for general required service time distributions). These yield the asymptotic behavior of W~(t) as t ~ oo and rather simple formulas in the exponential case. The unconditional average response times are also obtained.

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В этой статье мы изучаем вопрос об интегрируемости функций от момента первого попадания в компактные множества и функций от момента первого возвращения для случайных процессов с дискрет ным параметром. Мы рассматриваем сначала класс процессов с от рицательным сносом, принимающих значения в R+, и доказываем для них общие достаточные условия интегрируемости функций этих случайных моментов. Условия формулируются в «мартингальном духе», впервые предложенном Фостером, и обобщают соответству ющие результаты, полученные ранее. Во второй части статьи мы обращаемся к тому же вопросу для отраженных случайных блужда ний с нулевым сносом внутри области. Применяя результаты первой части, мы получаем условия интегрируемости некоторых функций от момента первого попадания и момента первого возвращения для отра женных случайных блужданий. Полученные оценки дают довольно тонкие результаты для первых упомянутых случайных моментов и дополняют соответствующие результаты в [1]. Наконец, мы выво дим границы для скорости сходимости переходных вероятностей эргодического отраженного случайного блуждания к соответствующей инвариантной мере. Ключевые слова и фразы: момент попадания, счетные цепи Мар кова, отраженное случайное блуждание с отражением на границе.

Let f Z n , n > 0g be an aperiodic irreducible recurrent (not necessarily positive recurrent) Markov chain taking values on a countable unbounded subset S of R d , ð( . ) its invariant measure and f is a non-negative function de®ned on S. We ®rst ®nd suf®cient conditions under which S f (z)ð(dz) I (the corresponding result for the ®niteness of S f (z)ð(dz) was obtained by Tweedie). Then we obtain lower and upper bounds for the values of the invariant measure ð on the subsets B of S, that is, ð(B). These bounds are expressed in terms of ®rst passage probabilities and the ®rst exit time from B. We also show how to estimate the latter quantities using sub-or supermartingale techniques. The results are ®nally illustrated for driftless re¯ected random walks in Z 2 and for Markov chains on nonnegative reals with asymptotically small drift of Lamperti type. In both cases we obtain very precise information on the asymptotic behaviour of their stationary measures.