Recently, Bartoszewicz [5,6] considered mixtures of exponential distributions treated as the Laplace transforms of mixing distributions and established some stochastic order relations between them: star order, dispersive order, dilation. In this article the preservation of the likelihood ratio, hazard rate, reversed hazard rate, mean residual life, and excess wealth orders under exponential mixtures is studied. Some new preservation results for the dispersive order are given, as well as the preservation of the convex transform order, and the star one is discussed.

Recently, a Pollaczek-Khintchine-like formulation for M/G/l queues with disasters has been obtained. A disaster is said to occur if a negative arrival causes all the customers (and therefore work) to depart from the system immediately. This study generalizes this result further, as it is shown to hold even when negative arrivals cause only part of the work to be demolished. In other words, an arbitrary amount of work, following a known distribution, is allowed to be removed at a negative event. Under these circumstances, a general approach for obtaining the Pollaczek–Khintchine Formula is proposed, which is then illustrated via several examples. Typically, it is seen that the formulainvolves certain parameters that are not explicitly known. The formula itself is made possible due to the number in system being geometric under preemptive last in-first out discipline.

We consider two nodes in tandem. At each node or service center, there are two exponential servers present with the same service rate μ and each with its own queue. Customers arrive at the first node according to a Poisson process with arrival rate λ. At their arrival, they have to be assigned to one of the servers, so they are routed to one of the queues at node 1. Customers leaving center 1 enter node 2 and are routed to one of the queues at node 2. In this paper, we consider the case with full information and the case with partial information.

A model for a system whose state changes continuously with time is introduced. It is assumed that the system is modeled by Brownian motion with negative drift and an absorbing barrier at the origin. A repairman arrives according to a Poisson process and increases the state of the system by a random amount if the state is below a threshold α. Explicit expressions are deduced for the distribution function of X(t), the state of the system at time 1, if X(t) ≤ α and for the Laplace transform of the density of X( t). The stationary case is examined in detail.

Multiclass single-server systems with significant setup times (polling models) are common in industry. This article considers asymptotics for polling models with increasing setup times. Two types of polling model are considered, namely (a) polling models with polling tables, exhaustive service, and deterministic setups, and (b) cyclic exhaustive service polling models with general setups under heavy traffic. It is shown that as the mean setup time increases to infinity, the scaled intervisit time for each queue (time between service of that queue) converges in probability to a constant. This, in turn, is shown to imply that scaled steady-state waiting time converges in distribution to either a uniform distribution or a simple discrete random variable multiplied by a uniform random variable as setups tend to infinity. These results lead to considerable insight into the behavior of systems with setups, and conclusions are drawn with respect to previous studies.

Previous studies on emergency management of large-scale urban networks have commonly concentrated on system development to off-load intensive computations to remote cloud servers or improving communication quality during a disaster and ignored the effect of energy consumption of vehicles, which can play a vital role in large-scale evacuation owing to the disruptions in energy supply. Hence, in this paper we propose a cloud-enabled navigation system to direct vehicles to safe areas in the aftermath of a disaster in an energy and time efficient fashion. A G-network model is employed to mimic the behaviors and interactions between individual vehicles and the navigation system, and analyze the effect of re-routing decisions toward the vehicles. A gradient descent optimization algorithm is used to gradually reduce the evacuation time and fuel consumption of vehicles by optimizing the probabilistic choices of linked road segments at each intersection. The re-routing decisions arrive at the intersections periodically and will expire after a short period. When a vehicle reaches an intersection, if the latest re-routing decision has not expired, the vehicle will follow this advice, otherwise, the vehicle will stick to the shortest path to its destination. The experimental results indicate that the proposed algorithm can reduce the evacuation time and the overall fuel utilization especially when the number of evacuated vehicles is large.

We consider an exponential queueing system with multiple stations, each of which has an infinite number of servers and a dedicated arrival stream of jobs. In addition, there is an arrival stream of jobs that choose a station based on the state of the system. In this paper we describe two heavy traffic approximations for the stationary joint probability mass function of the number of busy servers at each station. One of the approximations involves state-space collapse and is accurate for large traffic loads. The state-space in the second approximation does not collapse. It provides an accurate estimate of the stationary behavior of the system over a wide range of traffic loads.

We consider a continuous review (s, S) model of perishable items with lost sales. Once items are perished the entire inventory drops instantaneously to zero. The total cost includes the cost of: ordering, unsatisfied demand, units destroyed, holding, and fixed cost of perishability. Both the time to perishability and the lead times are assumed to be exponentially distributed while two cases of demand distribution are considered: Poisson and compound Poisson with general demand sizes. We study the average cost criterion and provide computational results on the problem of finding the optimal re-order level, s, and order up-to level, S. None of the known work on the subject is as general as the model presented here. Our analysis leads to several insights on the optimal (s, S) policies for perishable items in the presence of lead times. For example, we demonstrate that the effectiveness of a heuristic that ignores perishability (and is also analyzed here) decreases with the demand variability and that the cost may either increase or decrease with this variability.

The linear k-within-(r, s)-out-of-(m, n):F lattice system consists of mn components arranged in m rows and n columns and fails whenever there is at least one rectangle of dimension r × s which contains k or more failed components. We propose recursive formulas for the calculation of the linear k-within-(r, s)-out-of-(m, n):F lattice system. The computing complexity of the system reliability using the recursive formulas is O((n − s)ms+1 + m2s) for r = m and O((r + 1)rs(m−r+1)[(n − s)m + 2r(s−1)ms]) for r < m.

In this article, we study the asymptotic equipartition property (AEP) for asymptotic circular Markov chains. First, the definition of an asymptotic circular Markov chain is introduced. Then by applying the limit property for the bivariate functions of nonhomogeneous Markov chains, the strong limit theorem on the frequencies of occurrence of states for asymptotic circular Markov chains is established. Next, the strong law of large numbers on the frequencies of occurrence of states for asymptotic circular Markov chains is obtained. Finally, we prove the AEP for asymptotic circular Markov chains.

Two main methods are used to solve continuous-time quasi birth-and-death processes: matrix geometric (MG) and probability generating functions (PGFs). MG requires a numerical solution (via successive substitutions) of a matrix quadratic equation A0 + RA1 + R2A2 = 0. PGFs involve a row vector $\vec{G}(z)$ of unknown generating functions satisfying $H(z)\vec{G}{(z)^\textrm{T}} = \vec{b}{(z)^\textrm{T}},$ where the row vector $\vec{b}(z)$ contains unknown “boundary” probabilities calculated as functions of roots of the matrix H(z). We show that: (a) H(z) and $\vec{b}(z)$ can be explicitly expressed in terms of the triple A0, A1, and A2; (b) when each matrix of the triple is lower (or upper) triangular, then (i) R can be explicitly expressed in terms of roots of $\det [H(z)]$; and (ii) the stability condition is readily extracted.

Consider a finite-capacity telecommunications link to which connection requests arrive in a Poisson process. Each connection carried on the link earns a certain amount of revenue for the link's manager. Now, assume that the manager is offered the opportunity to buy or sell a unit of the link's allocated capacity. Assuming that the manager has a knowledge of the current number of connections on the link, we demonstrate a method of calculating the buying and selling prices.

In this paper, we consider a discrete-time queueing model for a Time Division Multiplexing (TDM) system with integration of voice and data (a model introduced by Li and Mark [16]). The voice traffic is a superposition of N Markov chains, which alternate between two states: the talkspurt state and the silence state. The data traffic is Poisson and independent of the voice sources. We show that the average queue size is increasing in certain correlation coefficients of the voice sources, increasing convex in the proportion of time the voice sources are in talkspurts, increasing convex in the number of voice sources, and increasing convex in the data traffic intensity. However, it is decreasing convex in the number of channels. These structural results yield various bounds. To take video traffic into account as well, we adapt a model of Maglaris et al. [18]. In their model, video traffic is generated by a continuous-state autoregressive Markov process that matches the average rate and the autocovariance of the output of a video coder. We show that if we replace their autoregressive model by a two-state Markov chain model with the same rate and correlation coefficient, we obtain an upper bound for the queue size. This replacement enables us to treat the video traffic as a voice source and use the techniques developed for dealing with voice/data integration to obtain bounds and estimates.

We consider tandem queueing systems with a general arrival process and exponential service distribution. The queueing system consists of several stations with finite intermediate buffer capacity between the stations. We address the problem of determining the optimal arrangement for the stations. We find that considering the last two stations, the departure process is stochastically faster if the slower station is last. Our results are consistent with the “bowl shape” phenomenon that has been observed in serial queueing systems with zero buffer capacity.

We consider the problem of dynamic allocation of a single server with batch processing capability to a set of parallel queues. Jobs from different classes cannot be processed together in the same batch. The arrival processes are mutually independent Poisson flows with equal rates. Batches have independent and identically distributed exponentially distributed service times, independent of the batch size and the arrival processes. It is shown that for the case of infinite buffers, allocating the server to the longest queue, stochastically maximizes the aggregate throughput of the system. For the case of equal-size finite buffers the same policy stochastically minimizes the loss of jobs due to buffer overflows. Finally, for the case of unequal-size buffers, a threshold-type policy is identified through an extensive simulation study and shown to consistently outperform other conventional policies. The good performance of the proposed threshold policy is confirmed in the heavy-traffic regime using a fluid model.

We analyze the optimal preemptive sequencing of n jobs on two machines to minimize expected total flow time. The running times of the jobs are independent samples from the distribution Pr(X = 1) = p, Pr(X = κ + 1) = 1 − p. We verify that the shortest-expected-remaining-processing-time (SERPT) policy, which is optimal for independent and identically distributed (i.i.d.) running times with a monotone hazard-rate distribution, is not optimal for this distribution. However, we prove that if p ≥ 1/κ, then the number of decisions where SERPT and an optimal policy disagree is bounded by a constant independent of n. For p < 1/k, we prove that the expected number of such decisions has a similar bound. In addition, bounds on the expected increase in flow times under SERPT are derived; these bounds are also independent of n.

In this paper we analyze a queueing network consisting of parallel queues and arriving customers that have to be assigned to one of the queues. The assignment rule may not depend on the numbers of customers in the queues. Our goal is to find a policy that is optimal with respect to the long-run average cost. We will consider two cases: holding costs and waiting times. A recently developed algorithm for Markov decision chains with partial state information is applied. It turns out that the periodic policies found by this algorithm are close, if not equal, to the optimal ones.

This paper deals with the MX/G/c queue. Using analytical results for the special cases of the MX/M/c queue and the MX/D/c queue, a two-moment approximation is proposed for the waiting-time percentiles in the general case. This approximation is based on a linear interpolation with respect to the squared coefficient of variation of the service time distribution. Validation experiments indicate that this approximation performs quite well for practical purposes. In particular, the practically important percentiles in the tail of the waiting-time distribution are approximated extremely well.

Generalized semi-Markov processes and stochastic Petri nets have been proposed as general frameworks for a discrete event simulation on a countable state space. The two formal systems differ, however, with respect to the clock setting (event scheduling) mechanism, the state transition mechanism, and the form of the state space. We obtain conditions under which the marking process of a stochastic Petri net “mimics” a generalized semi-Markov process in the sense that the two processes (and their underlying general state-space Markov chains) have the same finite dimensional distributions. The results imply that stochastic Petri nets have at least the modeling power of generalized semiMarkov processes for discrete event simulation.

The loss probabilities of customers in the Mx/GI/1/k, GI/Mx/l/k and their related queues such as server vacation models are compared with respect to the convex order of several characteristics, for example, batch size, of the arrival or service process. In the proof, we give a characterization of a truncation expression for a stationary distribution of a finite Markov chain, which is interesting in itself.