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We consider the patient-to-bed assignment problem that arises in hospitals. Both emergency patients who require hospital admission and elective patients who have had surgery need to be found a bed in the most appropriate ward. The patient-to-bed assignment problem arises when a bed request is made, but a bed in the most appropriate ward is unavailable. In this case, the next-best decision out of a many alternatives has to be made, according to some suitable decision making algorithm. We construct a Markov chain to model this problem in which we consider the effect on the length of stay of a patient whose treatment and recovery consists of several stages, and can be affected by stays in or transfers to less suitable wards. We formulate a dynamic program recursion to optimise an objective function and calculate the optimal decision variables, and discuss simulation techniques that are useful when the size of the problem is too large. We illustrate the theory with some numerical examples. [ABSTRACT FROM AUTHOR]

Stochastic fluid models have been widely used to model the level of a resource that changes over time, where the rate of variation depends on the state of some continuous-time Markov process. Latouche and Taylor (Queueing Syst 63:109–129, 2009) introduced an approach, using matrix analytic methods and the reduced load approximation for loss networks, to analyse networks of fluid models all driven by the same modulating process where the buffers are infinite. We extend the method to networks involving finite buffer models and illustrate the approach by deriving performance measures for a simple network as characteristics such as buffer size are varied. Our results provide insight into the situations where the infinite buffer model is a reasonable approximation to the finite buffer model. [ABSTRACT FROM AUTHOR]

We prove the conjecture formulated in Litvak and Ejov (2009), namely, that the trace of the fundamental matrix of a singularly perturbed Markov chain that corresponds to a stochastic policy feasible for a given graph is minimised at policies corresponding to Hamiltonian cycles. [ABSTRACT FROM AUTHOR]

In this paper we introduce a structure called the Markovian tree (MT). We define the MT and explore its alternative representation as a continuous-time Markovian Multitype Branching Process. We then develop two algorithms, the Depth and Order algorithms to determine the probability of eventual extinction of the MT process. We show that both of these algorithms have very natural physically intuitive interpretations and are analogues of the Neuts and U algorithms in Matrix-analytic Methods. Furthermore, we show that a special case of the Depth algorithm sheds new light on the interpretation of the sample paths of the Neuts algorithm. [ABSTRACT FROM AUTHOR]

AbstractWe consider a two-class priority queue in which the priority of a customer increases linearly at some constant, class-dependent rate. We describe the related maximum priority process {(M1(t),M2(t)):t≥0}, which is of interest since the stationary distribution of the maximum priority process at the times of the commencement of service gives information about the waiting time distributions of the two classes of customers. In the case where service times are exponential, we map a two-class maximum priority process {(M1(t),M2(t)):t≥0} to a tandem fluid queue, and use this mapping in order to derive the stationary distribution of {(M1(t),M2(t)):t≥0}. Further, we extended these results to the maximum priority process in which service times are phase-type distributed. We illustrate the theory with numerical examples. [ABSTRACT FROM AUTHOR]

Abstract Deriving the time-dependent expected reward function associated with a continuous-time Markov chain involves the computation of its transient deviation matrix. In this paper we focus on the special case of a finite quasi-birth-and-death (QBD) process, motivated by the desire to compute the expected revenue lost in a MAP/PH/1/C queue. We use two different approaches in this context. The first is based on the solution of a finite system of matrix difference equations; it provides an expression for the blocks of the expected reward vector, the deviation matrix, and the mean first passage time matrix. The second approach, based on some results in the perturbation theory of Markov chains, leads to a recursive method to compute the full deviation matrix of a finite QBD process. We compare the two approaches using some numerical examples. [ABSTRACT FROM AUTHOR]