Profit Analysis of the <Emphasis Type="Italic">M/M/R</Emphasis> Machine Repair Problem with Spares and Server Breakdowns

This paper studies the machine-repair problem consisting of M operating machines with S spares, and R servers which themselves are subject to breakdown under steady-state conditions. Spares are considered to be either cold-standby, or warm-standby or hot-standby. Failure and service times of the machines, and breakdown and repair times of the servers, are assumed to follow a negative exponential distribution. Each server is subject to breakdown even if no failed machines are in the system. A profit model is developed in order to determine the optimal values of the number of servers and spares. Numerical results are provided in which several system characteristics are evaluated for all cases under the optimal operating conditions.     

Optimal Control of an <Emphasis Type="Italic">M/E</Emphasis><Subscript><Emphasis Type="Italic">k</Emphasis></Subscript>/1 Queueing System with a Removable Service Station

This paper studies the optimal operation of an M/E _k /1 queueing system with a removable service station under steady-state conditions. Analytic closed-form solutions of the controllable M/E _k /1 queueing system are derived. This is a generalization of the controllable M/M/1, the ordinary M/E _k /1, and the ordinary M/M/1 queueing systems in the literature. We prove that the probability that the service station is busy in the steady-state is equal to the traffic intensity. Following the construction of the expected cost function per unit time, we determine the optimal operating policy at minimum cost.     

Optimizing System Availability Under Minimal Repair with Non-Negligible Repair and Replacement Times

Availability measures are given for a repairable system under minimal repair with constant repair times. A new policy and an existing replacement policy for this type of system are discussed. Each involves replacement at the first failure after time T, with T representing total operating time in the existing model and total elapsed time (i.e. operating time + repair time) in the new model. Optimal values of T are found for both policies over a wide range of parameter values. These results indicate that the new and administratively easier policy produces only marginally smaller optimal availability values than the existing policy.     

Mean First-passage Times for a <Emphasis Type="Italic">k</Emphasis>-out-of-<Emphasis Type="Italic">n</Emphasis> Sytem with Repair

Consider a k-out-of-n system where the lifetimes of the components are independent and identically distributed exponential (λ) random variables. Each component has its own repair facility, the repair times being independent and identically distributed exponential (μ) random variables, independent of the failure times. The mean operating time and mean repair time during the cycle between two successive breakdowns are found using renewal theory and the expression for the system availability. Using these, the mean first-passage times from any of the operating states of the system to the down state, and the mean first-passage times from any of the down states to the operating state are found recursively.     

The 〈G<Emphasis Type="Italic">/M/r/</Emphasis>FIFO〉 Machine-Interference Model with State-Dependent Speeds

This article treats a version of the multiple machine-interference problem with r operatives under FIFO repair discipline. The running times of machine i are supposed to be identically and arbitrarily distributed random variables with density function f _i (x), i = 1,…, n. The repair times of all machines are assumed to be identically and exponentially distributed random variables with mean 1/μ. The paper provides the main steady-state operational characteristics of the system when the running and repair speeds are dependent on the number of machines in working order.     

A Graphical Aid for the Initial Purchase of ‘Insurance Type’ Spares

Expensive renewable spares known as ‘insurance type’ spares are often a major concern in the design and setting up of industrial, commercial and military systems. These spares, though low in demand, are critical to the system's operation and their unavailability can lead to excessive downtime costs. Due to their nature, the (S-1, S) inventory control model provides an appropriate replenishment policy for this class of items, where S is the maximum number of spares in inventory. A (S-1, S) model with Exponential distribution of failure-free operating time at each of a finite number of machines and Exponential distribution of re-supply lead-time is developed. A graphical aid is presented which, for a given number of machines, indicates the range of the ratio {mean lead-time/mean failure-free operating time} for which a minimum S is required in order to satisfy a service level constraint on the service measure Pr[a spare is available at a machine stoppage due to part failure].     

The Transient Behaviour of a Cyclic Queue

This paper studies a cyclic queueing problem arising in civil engineering earthmoving projects. There are m excavators and n trucks which queue to be loaded by the excavators. The problem is therefore equivalent to a machine interference problem with m operators and n machines. The size of the haul fleet must be optimized, so as to minimize the cost per unit volume of earth moved. Graphs which show regions of optimal n in the parameter space are presented for the steady-state D/D/1, M/D/1 and E _k /M/1 models. The effect of operating the M/M/m system in short work shifts, so that the initial conditions and transient behaviour are important, is then analysed and quantified as a correction to the optimal number of trucks in the haul fleet from the steady-state solution.     

A Generalized Block Replacement Policy with Minimal Repair and General Random Repair Costs for a Multi-unit System

A generalization of the block replacement policy (BRP) is proposed and analysed for a multi-unit system which has the specific multivariate distribution. Under such a policy an operating system is preventively replaced at times kT (k = 1, 2, 3,...), as in the ordinary BRP, and the replacement of the failed system at failure is not mandatory; instead, a minimal repair to the component of the system can be made. The choice of these two possible actions is based on some random mechanism which is age-dependent. The cost of the ith minimal repair of the component at age y depends on the random part C(y) and the deterministic part C_i(y). The aim of the paper is to find the optimal block interval T which minimizes the long-run expected cost per unit time of the policy.     

Continuous and Discrete Age-Replacement Policies

This paper considers the combined continuous and discrete age-replacement policies when units deteriorate with age and use: a unit is replaced preventively before failure at time T of age or at number N of uses, whichever occurs first. The expected cost rate C (T,N) is derived, and both optimum time T* and number N* to minimize C (T,N) are discussed. There exist finite and unique T* and N* when the use occurs in a Poisson process under suitable conditions.     

Reliability Analysis of a One-Unit System with Unrepairable Spare Units and its Optimization Applications

It is assumed that a unit is either in operation or is in repair. When the main unit is under repair, spare units which cannot be repaired are used. In this system the following quantities are of interest: (i) The time distribution and the mean time to first-system failure, given that the n spare units are provided at time 0. (ii) The probability that the number of the failed spare units are equal to exactly n during the interval (0, t], and its expected number during the interval (0, t]. These quantities are derived by solving the renewal-type equations.Two optimization problems are discussed using the results obtained, viz.: (i) The expected cost of two systems, one with both a main unit and spare units and the other with only spare units is considered. (ii) A preventive maintenance policy of the main unit is considered in order to minimize the expected cost rate. Some policies of the two problems are discussed under suitable conditions. Numerical examples are also presented.     

Costing a Finite Minimal Repair Replacement Policy

In this paper an integral equation technique is used to evaluate the expected cost for the period (0, t] of a policy involving minimal repair at failure with replacement after N failures. This cost function provides an appropriate criterion to determine the optimal replacement number N* for a system required for use over a finite time horizon. In an example, it is shown that significant cost savings can be achieved using N* from the new finite time horizon model rather than the value predicted by the usual asymptotic model.     

The construction of infinite families of any κ-tight optimal and singular κ-tight optimal directed double loop networks

The double loop network (DLN) is a circulant digraph with n nodes and outdegree 2. It is an important topological structure of computer interconnection networks and has been widely used in the designing of local area networks and distributed systems. Given the number n of nodes, how to construct a DLN which has minimum diameter? This problem has attracted great attention. A related and longtime unsolved problem is for any given non-negative integer k, is there an infinite family of k-tight optimal DLN? In this paper, two main results are obtained (1) for any k ≥ 0, the infinite families of k-tight optimal DLN can be constructed, where the number n(k,e,c) of their nodes is a polynomial of degree 2 in e with integral coefficients containing a parameter c. (2) for any k ≥ 0,an infinite family of singular k-tight optimal DLN can be constructed.     

On the Relative Waiting Times in the GI/M/s and the GI/M/1 Queueing Systems

The expected steady-state waiting time, Wq(s), in a GI/M/s system with interarrival-time distribution H(·) is compared with the mean waiting time, Wq, in an "equivalent" system comprised of s separate GI/M/1 queues each fed by an interarrival-time distribution G(·) with mean arrival rate equal to 1/s times that of H(·). For H(·) assumed to be Exponential, Gamma or Deterministic three possible relationships between H(·) and G(·) are considered: G(·) can be of the "same type" as H(·); G(·) can be derived from H(·) by assigning new arrivals to the individual channels in a cyclic order; and G(·) may be obtained from H(·) by assigning customers probabilistically to the different queues. The limiting behaviour of the ratio R = Wq/Wq(s) is studied for the extreme values (1 and 0) of the common traffic intensity, ρ. Closed form results, which depend on the forms of H(·) and G(·) and on the relationships between them, are derived. It is shown that Wq is greater than Wq(s) by a factor of at least (s + 1)/2 when ρ approaches one, and that R is at least s(s!) when ρ tends to zero. In the latter case, however, R goes to infinity (!) in most cases treated. The results may be used to evaluate the effect on the waiting times when, for certain (non-queueing) reasons, it is needed to partition a group of s servers into several small groups.     

Two-stage flexible flow shop scheduling subject to fixed job sequences

This paper investigates the scheduling problem in a two-stage flexible flow shop, which consists of m stage-1 parallel dedicated machines and a stage-2 bottleneck machine, subject to the condition that n _l jobs per type l∈{1, …, m} are processed in a fixed sequence. Four regular performance metrics, including the total completion time, the maximum lateness, the total tardiness, and the number of tardy jobs, are considered. For each considered objective function, we aim to determine an optimal interleaving processing sequence of all jobs coupled with their starting times on the stage-2 bottleneck machine. The problem under study is proved to be strongly NP-hard. An O(m²Π_l=1 ^m n _l ²) dynamic programming algorithm coupled with numerical experiments is presented.     

The Generalized <Emphasis Type="Italic">M/G/k</Emphasis> Blocking System with Heterogeneous Servers

This paper studies the equilibrium behaviour of the generalized M/G/k blocking system with heterogeneous servers. Such a service system has k servers, each with a (possibly) different service time distribution, whose customers arrive in accordance with a Poisson process. They are served, unless all the servers are occupied. In this case they leave and they do not return later (i.e. they are ‘blocked’). Whenever there are n (n = 0, 1, 2,..., k) servers occupied, each arriving customer balks with probability 1 - f _{n +1}(f _{k +1} = 0) and each server works at a rate g _n . Among other things, a generalization of the Erlang B-formula is given and also it is shown that the equilibrium departure process from the system is Poisson.     

Repair Replacement Modelling over Finite Time Horizons

In this paper an integral equation approach is given for evaluating the expected cost of repair replacement policies over finite time horizons. An asymptotic estimate of this expected cost is also obtained. The policy involving imperfect repair on failure with replacement after N failures is taken as an illustrative example and optimal policies N* are found for both infinite and finite time horizons of use.     

Optimal Reliability Design of <Emphasis Type="Italic">K</Emphasis>-out-of-<Emphasis Type="Italic">N</Emphasis> Systems Subject to Two Kinds of Failure

K-out-of-N systems formed from N identical and independent components are considered in which the components can take two states: 0 (open) or 1 (closed) on command (e.g. Electromagnetic Relays and Solid State Switches). The components are subject to two kinds of failure on command: failure to open and failure to close. A K-out-of-N system is closed if and only if at least K of its components are closed. The system is considered open or closed depending on the states of its components. The optimum system is taken to be that system which maximizes the reliability. This paper finds the optimum K-out-of-N system given a fixed number of components.     

Optimal inspection intervals for safety systems with partial inspections

The introduction of International Standard IEC 61508 and its industry-specific derivatives sets demanding requirements for the definition and implementation of life-cycle strategies for safety systems. Compliance with the Standard is important for human safety and environmental perspectives as well as for potential adverse economic effects (eg, damage to critical downstream equipment or a clause for an insurance or warranty contract). This situation encourages the use of reliability models to attain the recommended safety integrity levels using credible assumptions. During the operation phase of the safety system life cycle, a key decision is the definition of an inspection programme, namely its frequency and the maintenance activities to be performed. These may vary from minimal checks to complete renewals. This work presents a model (which we called ρβ model) to find optimal inspection intervals for a safety system, considering that it degrades in time, even when it is inspected at regular intervals. Such situation occurs because most inspections are partial, that is, not all potential failure modes are observable through inspections. Possible reasons for this are the nature and the extent of the inspection, or potential risks generated by the inspection itself. The optimization criterion considered here is the mean overall availability A _o , but also taking into account the requirements for the safety availability A _s . We consider several conditions that ensure coherent modelling for these systems: sub-systems decomposition, k-out-of-n architectures, diagnostics coverage (observable/total amount of failure modes), dependent and independent failures, and non-negligible inspection times. The model requires an estimation for the coverage and dependent-failure ratios for each component, global failure rates, and inspection times. We illustrate its use through case studies and compare results with those obtained by applying previously published methodologies.     

Economic design of an adaptive <Emphasis Type="Italic">T</Emphasis><Superscript>2</Superscript> control chart

Varying three parameters of the Hotelling's T² control chart, namely, the sample size, the sampling interval length, and the action limit, results in a great improvement over the traditional T² control chart particularly for the small process shift. This paper presents the economic model for designing the variable parameters T² chart. In the economic design, a cost function is constructed, involving the cost of sampling and testing, the cost of false alarm, the cost to detect and remove the assignable cause, and the cost when the process is operating out-of-control. Also, a heuristic method of finding optimal values of adaptive design parameters for minimizing the cost function is presented. Variable and fixed parameters T² control charts are compared with respect to the expected loss per unit time. Furthermore, the effects of the initial sample number (used for estimating the parameters of F-distribution) upon the operating cost and adaptive design parameters are examined.     

Scheduling policies in the <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">G</Emphasis>/1 make-to-stock queue

In this paper, we analyse a production/inventory system modelled as an M/G/1 make-to-stock queue producing different products requiring different and general production times. We study different scheduling policies including the static first-come-first-served, preemptive and non-preemptive priority disciplines. For each static policy, we exploit the distributional Little's law to obtain the steady-state distribution of the number of customers in the system and then find the optimal inventory control policy and the cost. We additionally provide the conditions under which it is optimal to produce a product according to a make-to-order policy. We further extend the application area of a well-known dynamic scheduling heuristic, Myopic(T), for systems with non-exponential service times by permitting preemption. We compare the performance of the preemptive-Myopic(T) heuristic alongside that of the static preemptive-bμ rule against the optimal solution. The numerical study we have conducted demonstrates that the preemptive-Myopic(T) policy is superior between the two and yields costs very close to the optimal.