The calculation of the distribution of a linear combination of order statistics from random variables that are uniformly distributed is considered. A simple recursion to compute this distribution is presented that, unlike previous methods, is numerically stable and efficient. As such, this should be the algorithm of choice when the linear combination distribution needs to be obtained.

Markov reward models have been employed to obtain performability measures of computer and communication systems. In these models, a continuous time Markov chain is used to represent changes in the system structure, usually caused by faults and repairs of its components, and reward rates are assigned to states of the model to indicate some measure of accomplishment at each structure. A procedure to calculate numerically the distribution of the reward accumulated over a finite observation period is presented. The development is based solely on probabilistic arguments, and the final recursion is quite simple. The algorithm has a low computational cost in terms of model parameters. In fact, the number of operations is linear in a parameter that is smaller than the number of rewards, while the storage required is independent of the number of rewards. We also consider the calculation of the distribution of cumulative reward for models in which impulse based rewards are associated with transitions.

Consider a switching component in a packet-switching network, where messages from several incoming channels arrive and are routed to appropriate outgoing ports according to a service policy. One requirement in the design of such a system is to determine the buffer storage necessary at the input of each channel and the policy for serving these buffers that will prevent buffer overflow and the corresponding loss of messages. In this paper, a class of buffer service policies, called Least Time to Reach Bound (LTRB), is introduced that guarantees no overflow, and for which the buffer size required at each input channel is independent of the number of channels and their relative speeds. Further, the storage requirement is only twice the maximal length of a message in all cases, and as a consequence the class is shown to be optimal in the sense that any nonoverflowing policy requires at least as much storage as LTRB.

Polling systems have long been the subject of study and are of particular interest in the analysis of high-speed communications networks. There are many options for the scheduling policies that can be used at each polling station (exhaustive, gated, customer limited, etc.). In addition, one can impose an upper bound on the total service time delivered to customers at a station per server visit. In the most common case the upper bound is a constant for each polling station, and the resulting system model is not Markovian even when service times and interarrival times are exponential. In the paper, a comprehensive solution is developed for the major scheduling policies with time limits for each polling station. The approach is based on studying the embedded Markov chain defined at the sequence of epochs when the server arrives at each polling station. The computation of transition probabilities for the embedded chain requires transient analysis of the Markov process describing the system evolution between epochs. Uniformization methods are used to develop efficient algorithms for the transition probabilities and for system performance measures. Example problems are solved using the techniques developed to illustrate the utility of the results.>

A solution method is developed to analyze various scheduled maintenance policies for repairable computer systems. The analysis is applicable to systems with behavior (exclusive of the scheduled maintenance policy) that can be modeled by a continuous-time Markov process, and thus important characteristics can be included in the model. Furthermore, the assumption of perfect repair, which is unrealistic for most systems, is not used. Both transient and steady-state measures are obtained. The measures considered include expected availability, expected number of unscheduled repairs (repairs that are performed outside of the scheduled maintenance period), and the probability of an unscheduled repair. The solution approach is based on the randomization technique and possesses advantage such as numerical stability and ease of implementation.>

Repairable computer systems are considered, the availability behavior of which can be modeled as a homogeneous Markov process. The randomization method is used to calculate various measures over a finite observation period related to availability modeling of these systems. These measures include the distribution of the number of events of a certain type, the distribution of the length of time in a set of states, and the probability of a near-coincident fault. The method is then extended to calculate performability distributions. The method relies on coloring subintervals of the finite observation period based on the particular application, and then calculating the measure of interest using these colored intervals.

We consider repairable computer systems, those for which repair can be performed to put the system back in operation. The behavior of the system is assumed to be modeled as a homogeneous Markov process. We calculate numerically the distribution of cumulative operational time, which is the distribution of the total time during which the system was in operation over a finite observation period. The method is based on the randomization technique. The main advantages include the ability to specify error tolerances in advance, numerical stability, and simplicity of implementation. We also show that other quantities of interest can be calculated as a byproduct of the method without any significant extra computational effort.