H. Richard Gail

Calculating availability and performability measures of repairable computer systems using randomization

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.

Performability analysis of computer systems: from model specification to solution

Abstract Computer systems reliability/availability modeling deals with the representation of changes in the structure of the system being modeled, which are generally due to faults, and how such changes affect the availability of the system. On the other hand, performance modeling involves representing the probabilistic nature of user demands and predicting the system capacity to perform useful work, under the assumption that the system structure remains constant. With the advent of degradable systems, the system may be restructured in response to faults and may continue to perform useful work, even though operating at lower capacity. Performability modeling considers the effect of structural changes and their impact on the overall performance of the system. The complexity of current computer systems and the variety of different problems to be analyzed, including the simultaneous evaluation of performance and availability, demonstrate the need for sophisticated tools that allow the specification of general classes of problems while incorporating powerful analytic and/or simulation techniques. Concerning model specification, a recently proposed object oriented modeling paradigm that accommodates a wide variety of applications is discussed and compared with other approaches. With respect to solution methods, a brief overview of past work on performability evaluation of Markov models is presented. Then it is shown that many performability related measures can be calculated using the uniformization or randomization technique by coloring distinguished states and/or transitions of the Markov model of the system being studied. Finally, the state space explosion problem is addressed and several techniques for dealing with the problem are discussed.

An algorithm to calculate transient distributions of cumulative rate and impulse based reward

Markov reward models have been used to solve a wide variety of problems. In these models, reward rates are associated to the states of a continuous time Markov chain, and impulse rewards are associated to transitions of the chain. Rate based rewards are gained per unit time in the associated state, while impulse rewards are gained instantaneously each time certain transitions occur. We develop an efficient algorithm to calculate the distribution of the total accumulated reward over a given interval of time when both rate and impulse rewards are present. As special cases, we obtain an algorithm to handle models for which only rate rewards occur and another algorithm for the case when only impulse rewards are present. The development is based purely on probabilistic arguments, and the recursions obtained are simple and have a low computational cost.

Transient Solutions for Markov Chains

Much of the theory developed for solving Markov chain models is devoted to obtaining steady state measures, that is, measures for which the observation interval (0, t) is “sufficiently large” (t → ∞). These measures are indeed approximations of the behavior of the system for a finite, but long, time interval, where long means with respect to the interval of time between occurrences of events in the system. However, an increasing number of applications requires the calculation of measures during a relatively “short” period of time. These are the so-called transient measures. In these cases the steady state measures are not good approximations for the transient, and one has to resort to different techniques to obtain the desired quantities.

Calculating transient distributions of cumulative reward

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.

Use of Characteristic Roots for Solving Infinite State Markov Chains

In this chapter, our interest is in determining the stationary distribution of an irreducible positive recurrent Markov chain with an infinite state space. In particular, we consider the solution of such chains using roots or zeros. A root of an equation f (z) = 0 is a zero of the function f (z),and so for notational convenience we use the terms root and zero interchangeably. A natural class of chains that can be solved using roots are those with a transition matrix that has an almost Toeplitz structure. Specifically, the classes of M/G/1 type chains and G/M/1 type chains lend themselves to solution methods that utilize roots. In the M/G/1 case, it is natural to transform the stationary equations and solve for the stationary distribution using generating functions. However, in the G/M/1 case the stationary probability vector itself is given directly in terms of roots or zeros. Although our focus in this chapter is on the discrete-time case, we will show how the continuous-time case can be handled by the same techniques. The M/G/1 and G/M/1 classes can be solved using the matrix analytic method [Neuts, 1981, Neuts, 1989], and we will also discuss the relationship between the approach using roots and this method.

An optimal service policy for buffer systems

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.

Calculating the Distribution of a Linear Combination of Uniform Order Statistics

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.

Performance analysis of the IEEE 1394 serial bus

IEEE 1394 is a standard for a high performance serial bus interface. It encompasses both isochronous data transfer, which is suitable for real-time applications, and asynchronous data transfer, which is appropriate for delay-insensitive applications. This standard can be used as a basis for constructing a small-size local area network. Two queueing models are proposed for a network operating under the IEEE 1394 specification. The average waiting time in steady state of an asynchronous packet is calculated, and the effect on it due to isochronous traffic is studied. Numerical results from both analysis and simulation are presented in order to evaluate the performance of such a system.

Efficient Solutions for a Class of Non-Markovian Models

Although the use of embedded Markov chains has been known for some time, the application of this technique has been very ad hoc and has not been established as a standard approach for a wide class of models. Recently however, there has been progress in the direction of identifying an interesting class of models which are not Markovian but which can yield to a well defined solution method based on the analysis of an embedded Markov chain. Example applications that yield to this approach include polling models with deterministic timeout periods and models with deterministic service time queues. In this paper we derive efficient methods for computing both the transition probabilities for the embedded chain and performance measures expressible as Markov reward functions. Calculating the transition probabilities for the embedded chain requires transient analysis, and our computational procedures are based on uniformization. Examples are given to demonstrate the effectiveness of the methods and the extended class of models that are solvable with these techniques.