Bounds For Markovian Queues With Possible Catastrophes

We consider a general Markovian queueing model with possible catastrophes and obtain new and sharp bounds on the rate of convergence. Some special classes of such models are studied in details, namely, (a) the queueing system with S servers, batch arrivals and possible catastrophes and (b) the queueing model with “attracted” customers and possible catastrophes. A numerical example illustrates the calculations. Our approach can be used in modeling information flows related to high-performance computing. INTRODUCTION There is a large number of papers devoted to the research of Markovian queueing models with possible catastrophes, see for instance, [1], [3], [2], [10], [11], [17], [18], [19], [21], [24], [25] and the references therein. Such models are widely used in simulations for hight-performance computing. In particular, in some recent papers the authors deal with more or less special birth-death processes with additional transitions from and to origin [1], [2], [3], [10], [11], [21], [24], [25]. In the present paper we consider a more general class of Markovian queueing models with possible catastrophes and obtain key bounds on the rate of convergence, which allow us to compute the limiting characteristics of the corresponding processes. Namely, we suppose that the queue-length process is an inhomogeneous continuous-time Markov chain {X(t), t ≥ 0} on the state space E = {0, 1, 2 . . . }. All possible transition intensities are assumed to be non-random functions of time and may depend on the state of the process. From any state i the chain can jump to any another state j > 0 with transition intensity qij(t). Moreover, the transition functions from state i > 0 to state 0 (catastrophe intensities) are βi(t). Denote by pij (s, t) = P {X (t) = j |X (s) = i}, i, j ≥ 0, 0 ≤ s ≤ t the probability of transition X (t), and by pi (t) = P {X (t) = i} the corresponding state probability that X (t) is in state i at the moment t. Let p (t) = (p0 (t) , p1 (t) , . . . ) T be the vector of state probabilities at the moment t. Throughout the paper we suppose that for any i, j P (X (t+ h) = j|X (t) = i) = =  qij (t)h+ αij (t, h) , if j 6= i, βi (t)h+ αi0 (t, h) = qi0(t) + αi0(t, h), if j = 0, i > 1, 1− ∑ j 6=i qij(t)h+ αi (t, h) , if j = i, (1) where sup i |αi(t, h)| = o(h). (2) Let Q(t) be the corresponding intensity matrix. We suppose that all intensity functions are non-negative and locally integrable on [0,∞). Put aij (t) = qji (t) for j 6= i and aii (t) = − ∑