A deteriorating repairable system with stochastic lead time and replaceable repair facility

In this paper, a deteriorating repairable system with stochastic lead time and replaceable repair facility is studied. We assume that the spare system for replacement is available only by an order and the lead time for delivering the spare follows exponential distribution. Moreover, we also suppose that the repair facility may be subject to failure during the repair period. Under these assumptions, by using the geometric process and the supplementary variable technique, some important reliability indices such as the system availability, rate of occurrence of failure (ROCOF) and the probability that the system is waiting for replacement are derived. An ordering policy N − 1 and a replacement policy N based on the number of failures of the system are also considered. Furthermore, employing several Lemmas, the explicit expression of the average cost rate is derived. Meanwhile, the optimum value N^∗ for minimizing the average cost rate could be determined numerically.     

Information theoretic approximations for M/G/1 and G/G/1 queuing systems

Summary This paper presents new results concerning the use of information theoretic inference techniques in system modeling and concerning the widespread applicability of certain simple queuing theory formulas. For the case when an M/G/1 queue provides a reasonable system model but when information about the service time probability density is limited to knowledge of a few moments, entropy maximization and cross-entropy minimization are used to derive information theoretic approximations for various performance distributions such as queue length, waiting time, residence time, busy period, etc. Some of these approximations are shown to reduce to exact M/M/1 results when G = M. For the case when a G/G/1 queue provides a reasonable system model, but when information about the arrival and service distributions is limited to the average arrival and service rates, it is shown that various well known M/M/1 formulas are information theoretic approximations. These results not only provide a new method for approximating the performance distributions, but they help to explain the widespread applicability of the M/M/1 formulas.     

Optimal (N,T)-policy for M/G/1 system with cost structures

In this paper, the optimal (N,T)-policy for M/G/1 system with cost structure is studied. The system operates only intermittently. It is shut down when no customers are present. A fixed set-up cost of K>0 is incurred each time the system is reopened. Also, a holding cost of h>0 per unit time is incurred for each customer present. The (N,T)-policy studied for this system is as follows: the system reactivates as soon as N customers are present or the waiting time of the leading customer reaches a predefined time T (see A.S. Alfa, I. Frigui, Eur. J. Oper. Res. 88 (1996) 599-613; Y.N. Doganata, in: E. Arikan (Ed.), Communication, Control, and Signal Processing, 1990, pp. 1663–1669). Later on, as a comparison, the start of the timer count is relaxed as follows: the system reactivates as soon as N customers are present or the time units after the end of the last busy period reaches a predefined time T. For both cases, the explicit optimal policy (N^*,T^*) for minimizing the long-run average cost per unit time are obtained. As extreme cases, we include the simple optimal policies for N-and T-polices. Several counter-intuitive results are obtained about the optimal T-policies for both types of models.     

The optimal control of an M/G/1 queueing system with server vacations, startup and breakdowns

This paper studies the control policy of the N policy M/G/1 queue with server vacations, startup and breakdowns, where arrivals form a Poisson process and service times are generally distributed. The server is turned off and takes a vacation whenever the system is empty. If the number of customers waiting in the system at the instant of a vacation completion is less than N, the server will take another vacation. If the server returns from a vacation and finds at least N customers in the system, he requires a startup time before providing service until the system is again empty. It is assumed that the server breaks down according to a Poisson process and his repair time has a general distribution. The system characteristics of such a model are analyzed and the total expected cost function per unit time is developed to determine the optimal threshold of N at a minimum cost.     

An optimal replacement policy for repairable cold standby system with priority in use

In this article, a cold standby repairable system consisting of two nonidentical components and one repairman is studied. It is assumed that component 2 after a repair is “as good as new” while component 1 after a repair is not, but component 1 is given priority in use. Under these assumptions, by using the geometric process repair model, we consider a replacement policy N based on the number of failures of component 1 under which the system is replaced when the number of failures of component 1 reaches N. Our problem is to determine an optimal policy N* such that the long-run average cost per unit time (i.e. the average cost rate) of the system is minimized. The explicit expression of the average cost rate of the system is derived and the corresponding optimal replacement policy N* can be determined numerically. Finally, a special system with Weibull-distributed working time and repair time of component 1 is given to illustrate the theoretical results in this article.     

Locating an n-server facility in a stochastic environment

We consider a demand-responsive service system in which n mobile units (servers) are garaged at one facility. Service demands arrive in time as a homogenous Poisson process, but are located over the service region according to an arbitrary probability law. Given a random service demand, either (1) a mobile unit is dispatched to the demand's location to provide on-scene service or (2) the demand is lost (i.e. it is handled by some back-up system). The resultant queueing system is an M/G/n loss system operating in steady state. The objective is to locate the garage facility so that the average cost of response is minimized, where the cost of response is a weighted sum of mean travel time to a random serviced demand and the cost of a lost demand, the weights being the respective probabilities of occurrence. We show that the optimum facility location reduces to Hakimi's well-known minisum location.     

A geometric process repair model with inspections and its optimisation

In this article, a deteriorating simple repairable system with inspections, is studied. We assume that the system failure can only be detected by inspections and the repair of the system is not as good as new. Further assume that the successive working times of the system form a decreasing geometric process whereas the consecutive repair times form an increasing geometric process. Under these assumptions, we present a bivariate mixed policy (T,?N), respectively, based on the time interval between two successive inspections and the failure-number of the system. Our aim is to determine an optimal mixed policy (T,?N)* such that the long-run average cost per unit time (i.e. the average cost rate) is minimised. The explicit expression of the average cost rate is derived, and the corresponding optimal mixed policy can be determined numerically. Finally, we provide a numerical example to illustrate our model, and carry through some discussions and sensitivity analysis.     

Optimal replacement policy for a two-dissimilar-component cold standby system with different repair actions

In this paper, a cold standby repairable system consisting of two dissimilar components and one repairman is studied. When failures occur, the repair of both component 1 and component 2 are not ‘as good as new’. The consecutive operating times of component 1 after repair constitute a decreasing geometric process, while the repair times of component 1 are independent and identically distributed. For component 2, its failure is rectified by minimal repair, and the repair time is negligible. Component 1 has priority in use when both components are good. The replacement policy N is based on the failure number of component 1. Under policy N, we derive the explicit expression of the long-run average cost rate C(N) as well as the average number of repairs of component 2 before the system replaced. The optimal replacement policy N*, which minimises the long-run average cost rate C(N), is obtained theoretically. If the failure rate r(t) of component 2 is increasing, the existence and uniqueness of the optimal policy N* is also proved. Finally, a numerical example is given to validate the developed theoretical model. Some sensitivity analyses are provided to show the influence of some parameters, such as the costs for replacement and repair, and the parameters of the lifetime and repair time distributions of both components, to the optimal replacement policy N* and corresponding average cost rate C(N*).     

Randomized control of T-policy for an M/G/1 system

We study an M/G/1 queueing system with a server that can be switched on and off. The server can take a vacation time T after the system becomes empty. In this paper, we investigate a randomized policy to control a server with which, when the system is empty, the server can be switched off with probability p and take a vacation or left on with probability (1 − p) and continue to serve the arriving customers. For this system, we consider the operating cost and the holding cost where the operating cost consists of the system running and switching costs (start up and shut down costs). We describe the structure and characteristics of this policy and solve a constrained problem to minimize the average operating cost per unit time under the constraint for the holding cost per unit time.     

The BMAP/PH/N retrial queueing system operating in Markovian random environment

We consider the BMAP/PH/N retrial queueing system operating in a finite state space Markovian random environment. The stationary distribution of the system states is computed. The main performance measures of the system are derived. Presented numerical examples illustrate a poor quality of the approximation of the main performance measures of the system by means of the simpler queueing models. An effect of smoothing the traffic and an impact of intensity of retrials are shown.     

Three m-failure group maintenance models for M/M/N unreliable queuing service systems

This paper considers group maintenance problems for an unreliable service system with N independent operating servers and a Markovian queue. A specific class of group maintenance policies is developed where the repair is started as soon as the number of failed servers reaches a predetermined threshold. This is actually a Quasi Birth-and-Death Process with two dimensions, the level for the arrival/service process and the phase for the failure/repair process. Two models with positive repair time and another with instantaneous repair are considered. The matrix geometric approach is applied to calculate the steady state distribution and the expected average cost for all three models. For the theoretical analysis, this paper proves that there exists an optimal group maintenance parameter m^*, which can find the minimal average cost for all three models. Additionally, some mathematical properties and sensitivity analyses are numerically demonstrated based on various parameters. Finally, the comparisons of these three proposed models in many aspects are also discussed.     

Optimal management for infinite capacity N-policy M/G/1 queue with a removable service station

In this article, we consider an infinite capacity N-policy M/G/1 queueing system with a single removable server. Poisson arrivals and general distribution service times are assumed. The server is controllable that may be turned on at arrival epochs or off at service completion epochs. We apply a differential technique to study system sensitivity, which examines the effect of different system input parameters on the system. A cost model for infinite capacity queueing system under steady-state condition is developed, to determine the optimal management policy at minimum cost. Analytical results for sensitivity analysis are derived. We also provide extensive numerical computations to illustrate the analytical sensitivity properties obtained. Finally, an application example is presented to demonstrate how the model could be used in real applications to obtain the optimal management policy.     

Improved algorithms for dynamic lot sizing problems with incremental discount

We develop improved algorithms for the dynamic lot sizing problems with incremental discount, where the procurement cost is a concave piecewise linear function with m sections and the holding cost is linear. We decompose the problem carefully and present a new dynamic programming formulation. By using geometric techniques, we show that when m is fixed, the problem can be solved in O(T?log?T) time, and further O(T) time if the procurement cost is stationary.     

RNN / LSTM with modified Adam optimizer in deep learning approach for automobile spare parts demand forecasting

The spare parts demand forecasting is very much essential for the organizations to minimize the cost and prevent the stock outs. The demand of spare parts/ car sales distribution is an important factor in inventory control. The valuation of the demand is challenging as the automobile spare parts/car sales demand are often recurrent. The renowned empirical method adopts historical demand data to create the distribution of lead time demand. Although it works reasonably well when service requirements are relatively low, it has difficulty reaching high target service levels. In this paper, we proposed Recurrent Neural Networks/ Long-Short Term Memory (RNN / LSTM) with modified Adam optimizer to predict the demand for spare parts. In this LSTM, weight vectors are generated respectively. These weights are optimized using the Modified-Adam algorithm. The accuracy of the forecast and the performance of the inventory are considered in the experimental result. Experimental results confirm that RNN / LSTM with a Modified-Adam works well with minimal error compared to other existing methods. We conclude that the proposed RNN/LSTM with Modified-Adam algorithm is well suited for the prediction of automobile spare parts.

     

Relativized collapsing between BPP and PH under stringent oracle access

We propose a new model of stringent oracle access defined for a general complexity class. For example, when comparing the power of two machine models relative to some oracle set X, we restrict that machines of both types ask queries from the same segment of the set X. In particular, for investigating polynomial-time (or polynomial-size) computability, we propose polynomial stringency, bounding query length to any fixed polynomial of input length. Under such stringent oracle access, we show an oracle G such that BPP^G = PH^G.     

Number-dependent replacement model with random lead-time for systems under maintenance continuously

This article considers a number-dependent replacement policy where a system has two types of failures and is replaced at the nth type I failure (minor failure) or first type II failure (catastrophic failure), whichever occurs first where type I and type II failures are age dependent. Type I failures can be removed by restoration without any cost, since the maintenance work is executed continuously during the operation of the system. However, type II failure can be removed only through a replacement at a replacement cost. A spare unit for replacement can be delivered upon order and is available only when the random lead-time is finished. A model is developed for the average cost per unit time and is based on the stochastic behaviour of the assumed system and reflects the cost of storing a spare as well as system downtime. The optimal number for a minimum-cost policy is described and discussed. It is shown that the optimal number n* which minimises the cost rate is given by a unique solution of the equation under certain conditions.     

Optimal NT policies for M/G/1 system with a startup and unreliable server

This paper studies the control policies of an M/G/1 queueing system with a startup and unreliable server, in which the length of the vacation period is controlled either by the number of arrivals during the idle period, or by a timer. After all the customers are served in the queue exhaustively, the server immediately takes a vacation and operates two different policies: (i) the server reactivates as soon as the number of arrivals in the queue reaches to a predetermined threshold N or the waiting time of the leading customer reaches T units; and (ii) the server reactivates as soon as the number of arrivals in the queue reaches to a predetermined threshold N or T time units have elapsed since the end of the completion period. If the timer expires or the number of arrivals exceeds the threshold N, then the server reactivates and requires a startup time before providing the service until the system is empty. Furthermore, it is assumed that the server breaks down according to a Poisson process and his repair time has a general distribution. We analyze the system characteristics for each scheme. The total expected cost function per unit time is developed to determine the optimal thresholds of N and T at a minimum cost.     

Cost analysis of a finite capacity queue with server breakdowns and threshold-based recovery policy

This paper analyzes a repairable M/M/1/N queueing system under a threshold-based recovery policy. The threshold-based recovery policy means that the server breaks down only if there is at least one customer in the system, and the recovery can be performed when q (1 ≤ q ≤ N) or more customers are present. For this queueing system, a recursive method is applied to obtain steady-state solutions in neat closed-form expressions. We then develop some important system characteristics, such as the number of customers in the system, the probability that the server is busy, the effective arrival rate and the expected waiting time in the system, etc. A cost model is constructed to determine the optimal threshold value, the optimal system capacity and the optimal service rate at a minimum cost. In order to solve this optimization problem, the direct search method and Newton's method are employed. Sensitivity analysis is also conducted with various system parameters. Finally, we present some managerial insights through an application example.     

Optimal replacement policy for a general geometric process model with δ-shock

In this article, a general geometric process model with δ-shock for a deteriorating system and an improving system is studied. A system will fail as soon as a shock arrives with interarrival time being inside a real set. Assume that a replacement policy N is adopted by which the system will be replaced by a new, identical one at the time following the N-th failure. Then the optimal replacement policy N* for minimising the long-run average cost per unit time is determined analytically. Finally, the sensitivity analysis is studied.     

Applying foster’s criteria to a GI / PH /1 queuing system

Foster’s criteria are used to obtain necessary and sufficient conditions ensuring the recurrence and ergodicity of an embedded Markov chain for a GI / PH / 1 queuing system. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 148–156, May–June 2006.