Z. Khalil

Availability of Series Systems with Various Shut-off Rules

In series systems (reliability block diagram), failure of any component leads to system failure. However, depending on the shut-off rule, some components may continue to operate with the system down. In this paper we consider several shut-off rules for series systems performance. We calculate limiting system availability under these shut-off rules. In particular, availability results for 2-and 3-unit systems with constant failure and repair (hazard) rates are extended to systems of arbitrary size.

Some analytical results for congestion in subscriber line modules

In modern telephone exchanges, subscriber lines are usually connected to the so-called subscriber line modules. These modules serve both incoming and outgoing traffic. An important difference between these two types of calls lies in the fact that in the case of blocking due to all channels busy in the module, outgoing calls can be queued whereas incoming calls get busy signal and must be re-initiated in order to establish the required connection. The corresponding queueing model was discussed recently by Lederman, but only the model with losses has been studied analytically. In the present contribution, we study the model which takes into account subscriber retrials and investigate some of its properties such as existence of stationary regime, derive explicit formulas for the system characteristics, limit theorems for systems under high repetition intensity of blocked calls and limit theorems for systems under heavy traffic.

On the optimal total processing time using checkpoints

The authors investigate the problem of optimizing the expected blocking time duration by providing a schedule of checkpoints during the required job processing time. They give a general approach for determining the optimal checkpoint schedule and derive some cases when the optimal checkpointing is uniform. The model has applications in unreliable computing systems, multiclient computer service, data transmissions, etc. >

A characterization of probability distributions similar to the exponential

A concept of the lack-of-memory property at a given time point c > 0 is introduced. It is equivalent to the concept of the almost-lack-of-memory (ALM) property of the random variables. A representation theorem is given for the cumulative distribution function of such random variables as well as for corresponding decompositions in terms of independent random variables. It is shown that a periodic failure rate for a random variable is equivalent to the ALM property. In addition some properties of the service time of an unreliable server are observed.

On the reliability of a two-unit redundant system with random switchover time and two types of repair

Abstract The LS transform of the time to system failure distribution is derived for a two-unit system with general failure and general repair distributions. Two types of repair are considered. If the repair is made quickly as compared to the time to failure of the working unit then it is proved that the limit distribution of the time to failure is exponential.

On Standby Redundancy with Delayed Repair

This paper derives a) the Laplace-Stieltjes transform of the time-to-system-failure distribution, and b) the mean time-to-system-failure. The system consists of several elements with one repair facility which remains idle until a queue of failed units is built up.

Characterization of a general class of life-testing models

In this paper we establish a characterization theorem for a general class of life-testing models based on a relationship between conditional expectation and the failure rate function. As a simple application of the theorem, we characterize the gamma, Weibull, and Gompertz distributions, since they have many probabilistic and statistical properties useful in both biometry and engineering reliability.

Stochastic inequalities for M/G/1 retrial queues

Consider an M/G/1 retrial queue. The performance characteristics of such a system are available in explicit form; however they are cumbersome (these formulas include integrals of Laplace transform, solutions of functional equations, etc.) In this paper we use the general theory of stochastic orderings to investigate the monotonicity properties of the system relative to the strong stochastic ordering, convex ordering and Laplace ordering. These results imply in particular simple insensitive bounds for the stationary distribution of the number of customers in the system and the mean number of customers served during a busy period.

Performance analysis of a hybrid switching system where voice messages can be queued

In the classical model of a hybrid switching system with movable boundary it is assumed that blocked voice messages are lost and do not affect the further functioning of the system. We describe a more realistic model where blocked voice messages are queued and then are served once a channel becomes free. The main mathematical difficulty in the analysis of such models lies in the fact that the underlying stochastic process has as state space the whole quadrant ℤ+2. We reduce the problem to a set of equations defined over the lattice semi-strip {1,...,N} × ℤ+. This in turn allows us to use available general mathematical theories.

THE SERVICE TIME PROPERTIES OF AN UNRELIABLE SERVER CHARACTERIZE THE EXPONENTIAL DISTRIBUTION

Consider the total service time of a job on an unreliable server under preemptiverepeat-different and preemptive-resume service disciplines. With identical initial conditions, for both cases, we notice that the distributions of the total service time under these two disciplines coincide, when the original service time (without interruptions due to server failures) is exponential and independent of the server reliability. We show that this fact under varying server reliability is a characterization of the exponential distribution. Further we show, under the same initial conditions, that the coincidence of the mean values also leads to the same characterization.