A note on two IFR systems

We present probabilistic proofs for the following two facts: (i) A k out of n system of i.i.d (independent identically distributed). IFR (increasing failure rate) components has an IFR life distribution. (ii) A compound Poisson process with nonnegative i.i.d jumps with PF₂ distribution is IFR.     

Testing goodness of fit to the increasing failure rate family

One branch of the reliability literature is concerned with devising statistical procedures with various nonparametric “restricted family” model assumptions because of the potential improved operating characteristics of such procedures over totally nonparametric ones. In the single-sample problem with unknown increasing failure rate (IFR) distribution F, (1) maximum-likelihood estimators of F have been calculated, (2) upper or lower tolerance limits for F have been determined, and (3) tests of the null hypothesis that F is exponential have been constructed. Barlow and Campo proposed graphical methods for assessing goodness of fit to the IFR model when the validity of this assumption is unknown. This article proposes several analytic tests of the IFR null hypothesis based on the maximum distance and area between the cumulative hazard function and its greatest convex minorant (GCM), and the maximum distance and area between the total time on test statistic and its GCM. A table of critical points is provided to implement a specific test having good overall power properties.     

A cumulative damage shock model with imperfect preventive maintenance

In this article we consider a cumulative damage shock model under a periodic preventive maintenance (PM) policy. The PM is imperfect in the sense that each PM reduces the damage level by 100(1 – b)%, 0 < b < 1. A system suffers damage due to shocks and fails when the damage level exceeds some threshold. We derive a sufficient condition for the time to failure to have an IFR distribution. We also discuss the associated problem of finding the number of PM's that minimizes the expected cost rate.     

Finite Markov chain models skip-free in one direction

Finite Markov processes are considered, with bidimensional state space, such that transitions from state (n, i) to state (m, j) are possible only if m ≤ n + 1. The analysis leads to efficient computational algorithms, to determine the stationary probability distribution, and moments of first passage times.     

The asymptotic joint distribution of the state occupancy times in an alternating renewal process

An alternating renewal process starts at time zero and visits states 1,2,…,r, 1,2, …,r 1,2, …,r, … in sucession. The time spent in state i during any cycle has cumulative distribution function F_i, and the sojourn times in each state are mutually independent, positive and nondegenerate random variables. In the fixed time interval [0,T], let U_i(T) denote the total amount of time spent in state i. In this note, a central limit theorem is proved for the random vector (U_i(T), 1 ≤ i ≤ r) (properly normed and centered) as T → ∞.     

Peak rate of occurrence of a poisson process

Suppose that a nonhomogeneous Poisson process is observed for a length of time T, say Let λ (t) denote the mean value function of the process. It is assumed that λ (t) is first increasing then decreasing inside the interval (0, T) with peak at t = t₀, say. Three methods are given for estimating to. One of these methods is nonparametric, and the other two methods are based on the standard regression technique and the maximum likelihood principle The given resull has application in a problem of determining the azimuth of a target from the radar-impulse data. The time series of incoming signals may be approximated by the occurrence of a nonhomogeneous Poisson process with mean value function λ (t). The azimuth of the target is reasonably determined from the direction of the axis of the radar beam at the instant to, corresponding to the peak value of λ (t).     

Steady-state analysis of load balancing with Coxian-2 distributed service times

This paper studies load balancing for many-server (N servers) systems. Each server has a buffer of size b ? 1, and can have at most one job in service and b ? 1 jobs in the buffer. The service time of a job follows the Coxian-2 distribution. We focus on steady-state performance of load balancing policies in the heavy traffic regime such that the normalized load of system is λ = 1 ? N^?α for 0 < α < 0.5. We identify a set of policies that achieve asymptotic zero waiting. The set of policies include several classical policies such as join-the-shortest-queue (JSQ), join-the-idle-queue (JIQ), idle-one-first (I1F) and power-of-d-choices (Po d) with d = O(N^α log N). The proof of the main result is based on Stein's method and state space collapse. A key technical contribution of this paper is the iterative state space collapse approach that leads to a simple generator approximation when applying Stein's method.     

Using a birth‐and‐death process to estimate the steady‐state distribution of a periodic queue

If the number of customers in a queueing system as a function of time has a proper limiting steady‐state distribution, then that steady‐state distribution can be estimated from system data by fitting a general stationary birth‐and‐death (BD) process model to the data and solving for its steady‐state distribution using the familiar local‐balance steady‐state equation for BD processes, even if the actual process is not a BD process. We show that this indirect way to estimate the steady‐state distribution can be effective for periodic queues, because the fitted birth and death rates often have special structure allowing them to be estimated efficiently by fitting parametric functions with only a few parameters, for example, 2. We focus on the multiserver M_t/GI/s queue with a nonhomogeneous Poisson arrival process having a periodic time‐varying rate function. We establish properties of its steady‐state distribution and fitted BD rates. We also show that the fitted BD rates can be a useful diagnostic tool to see if an M_t/GI/s model is appropriate for a complex queueing system. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 664–685, 2015     

Preservation of unimodality under shock models

A number of results pertaining to preservation of aging properties (IFR, IFRA etc.) under various shock models are available in the literature. Our aim in this paper is to examine in the same spirit, the preservation of unimodality under various shock models. For example, it is proved that in a non-homogeneous Poisson shock model if {p_k}_K≥0, the sequence of probabilities with which the device fails on the kth shock, is unimodal then under some suitable conditions on the mean value function Λ (t), the corresponding survival function is also unimodal. The other shock models under which the preservation of unimodality is considered in this paper are pure birth shock model and a more general shock model in which shocks occur according to a general counting process. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 952–957, 1999     

Stochastic control of queueing systems

Suppose that the state of a queueing system is described by a Markov process { Y_t, t ≥ 0}, and the profit from operating it up to a time t is given by the function f(Y_t). We operate the system up to a time T, where the random variable T is a stopping time for the process Yt. Optimal stochastic control is achieved by choosing the stopping time T that maximizes Ef(Y_T) over a given class of stopping times. In this paper a theory of stochastic control is developed for a single server queue with Poisson arrivals and general service times.     

The single server queue in discrete time-numerical analysis IV

The nonlinear difference equation for the distribution of the busy period for an unbounded discrete time queue of M|G| 1 type is solved numerically by a monotone iterative procedure. A starting solution is found by computing a first passage time distribution in a truncated version of the queue.     

Exact expected values of variance estimators for simulation

We formulate exact expressions for the expected values of selected estimators of the variance parameter (that is, the sum of covariances at all lags) of a steady‐state simulation output process. Given in terms of the autocovariance function of the process, these expressions are derived for variance estimators based on the simulation analysis methods of nonoverlapping batch means, overlapping batch means, and standardized time series. Comparing estimator performance in a first‐order autoregressive process and the M/M/1 queue‐waiting‐time process, we find that certain standardized time series estimators outperform their competitors as the sample size becomes large. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Checking policy for deteriorating multistate systems

In this article a multistate system under some checking policy is considered. The system has n + 1 states: 0,1, …,n, and deteriorates gradually. State 0 is a normal (full capacity) state and states 1, …,n are considered unsatisfactory. Transition from state 0 to state 1 is considered a system failure. This failure can be detected only through checking, which entails a fixed cost c. The holding time in the undiscovered state i (i = 1, …,n) results in cost d_i per unit of time. For such a system, the algorithm of determining optimum checking times is given.     

IFR results for repairable systems

Consider a k-out-of-n system with independent repairable components. Assume that the repair and failure distributions are exponential with parameters {μ₁, ?,μ_n} and {λ₁, ?,λ_n}, respectively. In this article we show that if λ_i – μ_i = Δ for all i then the life distribution of the system is increasing failure rate (IFR).     

An integral equation for the second moment function of a geometric process and its numerical solution

In this article, an integral equation satisfied by the second moment function M₂(t) of a geometric process is obtained. The numerical method based on the trapezoidal integration rule proposed by Tang and Lam for the geometric function M(t) is adapted to solve this integral equation. To illustrate the numerical method, the first interarrival time is assumed to be one of four common lifetime distributions, namely, exponential, gamma, Weibull, and lognormal. In addition to this method, a power series expansion is derived using the integral equation for the second moment function M₂(t), when the first interarrival time has an exponential distribution.     

Hybrid minimal repair and age replacement maintenance policies,with nonnegligible repair times

A system undergoes minimal repair during [0, T] with a failure replacement on first failure during [T, a], or a planned replacement if the system is still functioning at elapsed time a. Repairs and replacements are not necessarily instantaneous. An expression is obtained for the asymptotic expected cost rate, and sufficient conditions are obtained for the optimum T* > 0. Several special cases are considered. A numerical investigation for a Weibull distributed time to first failure compares this elapsed-time policy with replacement on failure only, and also a policy based on system operating time or age. It is found that in many cases the elapsed-time-based policy is only marginally worse than one based on system age, and may therefore be preferred in view of its administrative convenience. © 1994 John Wiley & Sons, Inc.     

Optimal admission pricing policies for M/Ek/1 queues

This paper extends the Low-Lippman M/M/1 model to the case of Gamma service times. Specifically, we have a queue in which arrivals are Poisson, service time is Gamma-distributed, and the arrival rate to the system is subject to setting an admission fee p. The arrival rate λ(p) is non-increasing in p. We prove that the optimal admission fee p^* is a non-decreasing function of the customer work load on the server. The proof is for an infinite capacity queue and holds for the infinite horizon continuous time Markov decision process. In the special case of exponential service time, we extend the Low-Lippman model to include a state-dependent service rate and service cost structure (for finite or infinite time horizon and queue capacity). Relatively recent dynamic programming techniques are employed throughout the paper. Due to the large class of functions represented by the Gamma family, the extension is of interest and utility.     

A load-sharing model: The linear breakdown rule

An n-component parallel system is subjected to a known load program. As time passes, components fail in a random manner, which depends on their individual load histories. At any time, the surviving components share the total load according to some rule. The system's life distribution is studied under the linear breakdown rule and it is shown that if the load program is increasing, the system lifetime is IFR. Using the notion of Schur convexity, a stochastic comparison of different systems is obtained. It is also shown that the system failure time is asymptotically normally distributed as the number of components grows large. All these results hold under various load-sharing rules; in fact, we show that the system lifetime distribution is invariant under different load-sharing rules.     

Optimal admission pricing and service rate control of an M[x]/M/s queue with reneging

In this article we consider the optimal control of an M^[X]/M/s queue, s ≧ 1. In addition to Poisson bulk arrivals we incorporate a reneging function. Subject to control are an admission price p and the service rate μ. Thus, through p, balking response is induced. When i customers are present a cost h(i,μ,p) per unit time is incurred, discounted continuously. Formulated as a continuous time Markov decision process, conditions are given under which the optimal admission price and optimal service rate are each nondecreasing functions of i. In Section 4 we indicate how the infinite state space may be truncated to a finite state space for computational purposes.     

An alternative proof of the IFRA property of some shock models

Let , where A (t)/t is nondecreasing in t, {P(k)^1/k} is nonincreasing. It is known that H(t) = 1 — H (t) is an increasing failure rate on the average (IFRA) distribution. A proof based on the IFRA closure theorem is given. H(t) is the distribution of life for systems undergoing shocks occurring according to a Poisson process where P (k) is the probability that the system survives k shocks. The proof given herein shows there is an underlying connection between such models and monotone systems of independent components that explains the IFRA life distribution occurring in both models.