An optimal age maintenance for an M/G/1 queueing system

This paper discusses an optimal age maintenance scheme for a queueing system. Customers arrive at the system according to a Poisson process. They form a single queue and are served by a server with general service distribution. The system fails after a random time and corrective maintenance is performed at the failure. A preventive maintenance is also performed if the system is empty at age T where ‘age’ refers to the elapsed time since the previous maintenance was completed. If the system is not empty at age T, the system is used until it fails. At the failure, the customers in the system are lost and the arriving customers during the maintenance are also lost. By renewal theory, we study the optimal value of T which minimizes the average number of lost customers over an infinite time horizon.     

Optimal replacement and allocation of multi‐state elements in k‐within‐m‐from‐r/n sliding window systems

This paper proposes a new model that generalizes the linear sliding window system to the case of multiple failures. The considered k ‐within‐ m ‐from‐ r / n sliding window system consists of n linearly ordered multi‐state elements and fails if at least k groups out of m consecutive groups of r consecutive multi‐state elements have cumulative performance lower than the demand W . A reliability evaluation algorithm is suggested for the proposed system. In order to increase the system availability, maintenance actions can be performed, and the elements can be optimally allocated. A joint element allocation and maintenance optimization model is formulated with the objective of minimizing the total maintenance cost subjected to the pre‐specified system availability requirement. Basic procedures of genetic algorithms are adapted to solve the optimization problem. Numerical experiments are presented to illustrate the applications. Copyright © 2015 John Wiley & Sons, Ltd.     

AnM/M/s-consistent diffusion model for theGI/G/s queue

This paper develops a diffusion-approximation model for a stableGI/G/s queue: The queue-length process in theGI/G/s queue is approximated by a diffusion process on the nonnegative real line. Some heuristics on the state space and the infinitesimal parameters of the approximating diffusion process are introduced to obtain an approximation formula for the steady-state queue-length distribution. It is shown that the formula is consistent with the exact results for theM/M/s andM/G/ queues. The accuracy of the approximations for principal congestion measures are numerically examined for some particular cases.     

Continuity of the M/G/c queue

Consider an M/G/c queue with homogeneous servers and service time distribution F. It is shown that an approximation of the service time distribution F by stochastically smaller distributions, say F _n , leads to an approximation of the stationary distribution π of the original M/G/c queue by the stationary distributions π _n of the M/G/c queues with service time distributions F _n . Here all approximations are in weak convergence. The argument is based on a representation of M/G/c queues in terms of piecewise deterministic Markov processes as well as some coupling methods.      

On the Transient Behavior of the M/M/1 and M/M/1/M Queues

This paper gives a transient analysis of the classic M/M/1 and M/M/1/K queues. Our results are asymptotic as time and queue length become simultaneously large for the infinite capacity queue, and as the system’s storage capacity K becomes large for the finite capacity queue. We give asymptotic expansions for p_n(t), which is the probability that the system contains n customers at time t. We treat several cases of initial conditions and different traffic intensities. The results are based on (i) asymptotic expansion of an exact integral representation for p_n(t) and (ii) applying the ray method to a scaled form of the forward Kolmogorov equation which describes the time evolution of p_n(t).     

A value-adding approach to reliability under preventive maintenance costs and its applications

No equipment (system) can be perfectly reliable in spite of the utmost care and best efforts on the part of the designer, decision-maker and manufacturer. The two sides of maintenance are corrective and preventive maintenance. It is generally assumed that a preventive maintenance action is less costly than a repair maintenance action. We examine this proposition in detail on the basis of a failure-time model that relates conformance quality to reliability. Illustratively, we present reliability in the context of contracts with asymmetric information. The model shows how to overcome information rents through price distortions and quantity rationing. The paper ends with a conclusion and an outlook to future studies.     

Fluid queue driven by aGI/GI/1 queue

Consider a model consisting of two phases: the GI/GI/1 queue and a buffer which is fed by a fluid arriving from a single-server queue. The fluid output from the GI/GI/1 queue is of the on/off type with on- and off-periods distributed as successive busy and idle periods in the GI/GI/1 queue. The fluid pours out of the buffer at a constant rate. The steady-state performance of this model is studied. We derive the Laplace-Stieltjes transform of the stationary distribution function of the buffer content in the case of the M/GI/1 queue in the first phase. It is shown that this distribution depends on the form of the service-time distribution. Therefore, the replacement of an M/GI/1 queue by an M/M/1 queue is not correct, in general. Continuity estimates are derived in the cast where the buffer is fed from the GI/GI/1 queue. Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow Russia, 1996, Part II.     

Generalized M/G/C/C state dependent queueing models and pedestrian traffic flows

The generality and usefulness ofM/G/C/C state dependent queueing models for modelling pedestrian traffic flows is explored in this paper. We demonstrate that the departure process and the reversed process of these generalizedM/G/C/C queues is a Poisson process and that the limiting distribution of the number of customers in the queue depends onG only through its mean. Consequently, the models developed in this paper are useful not only for the analysis of pedestrian traffic flows, but also for the design of the physical systems accommodating these flows. We demonstrate how theM/G/C/C state dependent model is incorporated into the modelling of large scale facilities where the blocking probabilities in the links of the network can be controlled. Finally, extensions of this work to queueing network applications where blocking cannot be controlled are also presented, and we examine an approximation technique based on the expansion method for incorporating theseM/G/C/C queues in series, merge, and splitting topologies of these networks.     

Cyclotomic numbers and primitive idempotents in the ring GF(q)[x]/(x−1)

Let q be an odd prime power and p be an odd prime with gcd(p,q)=1. Let order of q modulo p be f, and q^f=1+pλ. Here expressions for all the primitive idempotents in the ring R_pⁿ=GF(q)[x]/(x^pn−1), for any positive integer n, are obtained in terms of cyclotomic numbers, provided p does not divide λ if n2. The dimension, generating polynomials and minimum distances of minimal cyclic codes of length pⁿ over GF(q) are also discussed.     

Continuity theorems for the M/M/1/n queueing system

In this paper continuity theorems are established for the number of losses during a busy period of the M/M/1/n queue. We consider an M/GI/1/n queueing system where the service time probability distribution, slightly different in a certain sense from the exponential distribution, is approximated by that exponential distribution. Continuity theorems are obtained in the form of one or two-sided stochastic inequalities. The paper shows how the bounds of these inequalities are changed if further assumptions, associated with specific properties of the service time distribution (precisely described in the paper), are made. Specifically, some parametric families of service time distributions are discussed, and the paper establishes uniform estimates (given for all possible values of the parameter) and local estimates (where the parameter is fixed and takes only the given value). The analysis of the paper is based on the level crossing approach and some characterization properties of the exponential distribution. Dedicated to Vladimir Mikhailovich Zolotarev, Victor Makarovich Kruglov, and to the memory of Vladimir Vyacheslavovich Kalashnikov.     

M/G/1/N vacation model with varying E-limited service discipline

We define and analyze anM/G/1/N vacation model that uses a service discipline that we call theE-limited with limit variation discipline. According to this discipline, the server provides service until either the system is emptied (i.e. exhausted) or a randomly chosen limit ofl customers has been served. The server then goes on a vacation before returning to service the queue again. The queue length distribution and the Laplace-Stieltjes transforms of the waiting time, busy period and cycle time distributions are found. Further, an expression for the mean waiting time is developed. Several previously analyzed service disciplines, including Bernoulli scheduling, nonexhaustive service and limited service, are special cases of the general varying limit discipline that is analyzed in this paper.     

Quasi-birth-and-death Markov processes with a tree structure and the MMAP[K]/PH[K]/N/LCFS non-preemptive queue

This paper studies a multi-server queueing system with multiple types of customers and last-come-first-served (LCFS) non-preemptive service discipline. First, a quasi-birth-and-death (QBD) Markov process with a tree structure is defined and some classical results of QBD Markov processes are generalized. Second, the MMAP[K]/PH[K]/N/LCFS non-preemptive queue is introduced. Using results of the QBD Markov process with a tree structure, explicit formulas are derived and an efficient algorithm is developed for computing the stationary distribution of queue strings. Numerical examples are presented to show the impact of the correlation and the pattern of the arrival process on the queueing process of each type of customer.     

On inference concerning time-dependent queue performance: The M/G/1 example

This paper proposes easily-computed approximations to the finite-time expected waiting time for anM/G/1 system starting from an empty state. Both unsaturated (ρ<1) and saturated (ρ>1) conditions are considered. Numerical evidence is presented to indicate that the quality of the approximations is usefully good, especially when ease of computation is an issue. Further, the methodology is adapted to assess expected waiting time when inference must be made from a random sample of service times, and the decision is made to do so nonparametrically, i.e., without fitting a specific function. The results appear reasonable and potentially useful, and are not burdensome to obtain. The methodology investigated can also be applied to the variety of queueing models that are close siblings ofM/G/1: priority and breakdowns and “vacations” being examples. Of course other approximating and inferential options remain to be investigated.     

The MoravaK-theory of algebraicK-theory spectra

Ifn2 the MoravaK-theoryK(n) _* of an algebraicK-theory spectrumKX vanishes for any ring or schemeX. This is proved using thev _n -complexes of Hopkins and Smith, together with the following theorem. The natural mapf:Q ₀S⁰BGL⁺ factors through the space ImJ. In particularf _*: _* ^s K _* annihilates CokerJ. These results are closely related to the Lichtenbaum-Quillen conjectures.Partially supported by an NSF grant.     

Optimal Operating Policy for an M/G/1 Exhaustive Server-Vacation Model

We consider an M/G/1 queueing system controlled by an exhaustive server–vacation policy, i.e, the server is turned off whenever the system becomes empty and it is turned on after a random time with at least a customer present in the system. In this paper, it is proved that there exists an exhaustive optimal policy which is of the form X + a(T - X)⁺, where, starting with the server off, X represents the time for the first arrival and T and a are non-negative real numbers. Using a classical average cost structure, the optimization problem is treated under the asymptotic average criterion. A structured definition of exhaustive policy is also derived.     

Note on an improved branch-and-bound algorithm to solve n/m/P/Fmax problems

The LOMNICKI algorithm for n/m/P/F_max flow-shop problems is unsuitable for large values ofn andm because of the time and size of storage required to attain an optimal solution. The form of presentation of the problem to the algorithm can influence its performance. The algorithm performance can be improved applying the algorithm to the problem and to its inverse at the same time, sharing both applications the best value and the best bound found. Further exploitation of proprieties of the inverse problem are useful also for solve hard instances of the problem.     

Entropy of C(K)-Valued Operators

We investigate how the entropy numbers (e_n(T)) of an arbitrary Hölder-continuous operator T: E→C(K) are influenced by the entropy numbers (_n(K)) of the underlying compact metric space K and the geometry of E. We derive diverse universal inequalities relating finitely many _n(K)'s with finitely many e_n(T)'s which yield statements about the asymptotically optimal behaviour of the sequence (e_n(T)) in terms of the sequence (_n(K)). As an application we present new methods for estimating the entropy numbers of a precompact and convex subset in a Banach space E, provided that the entropy numbers of its extremal points are known.     

On the Moments of Overflow and Freed Carried Traffic for the GI/M/C/0 System

In circuit-switched networks call streams are characterized by their mean and peakedness (two-moment method). The GI/M/C/0 system is used to model a single link, where the GI-stream is determined by fitting moments appropriately. For the moments of the overflow traffic of a GI/M/C/0 system there are efficient numerical algorithms available. However, for the moments of the freed carried traffic, defined as the moments of a virtual link of infinite capacity to which the process of calls accepted by the link (carried arrival process) is virtually directed and where the virtual calls get fresh exponential i.i.d. holding times, only complex numerical algorithms are available. This is the reason why the concept of the freed carried traffic is not used. The main result of this paper is a numerically stable and efficient algorithm for computing the moments of freed carried traffic, in particular an explicit formula for its peakedness. This result offers a unified handling of both overflow and carried traffics in networks. Furthermore, some refined characteristics for the overflow and freed carried streams are derived.     

Asymptotic behaviour of the loss probability of theM/G/1/K andG/M/1/K queues

This paper investigates the asymptotic behaviour of the loss probability of theM / G/1/K and G/M/1/K queues as the buffer size increases. It is shown that the loss probability approaches its limiting value, which depends on the offered load, with an exponential decay in essentially all cases. The value of the decay rate can be easily computed from the main queue parameters. Moreover, the close relation existing between the loss behaviour of the two examined queueing systems is highlighted and a duality concept is introduced. Finally some numerical examples are given to illustrate on the usefulness of the asymptotic approximation.     

The Atiyah-Segal completion theorem for C*-algebras

We generalize the Atiyah-Segal completion theorem to C ^*-algebras as follows. Let A be a C ^*-algebra with a continuous action of the compact Lie group G. If K _* ^G (A) is finitely generated as an R(G)-module, or under other suitable restrictions, then the I(G)-adic completion K _* ^G (A) is isomorphic to RK _*([A C(EG)]^G), where RK _* is representable K-theory for - C ^*-algebras and EG is a classifying space for G. As a corollary, we show that if and are homotopic actions of G, and if K _*(C ^* (G,A,)) and K _*(C ^* (G,A,)) are finitely generated, then K _*(C ^*(G,A,))K_*(C ^* (G,A,)). We give examples to show that this isomorphism fails without the completions. However, we prove that this isomorphism does hold without the completions if the homotopy is required to be norm continuous.This work was partially supported by an NSF Graduate Fellowship and by an NSF Postdoctoral Fellowship.