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In this paper we consider an inventory system with increasing concave ordering cost and average cost optimization criterion. The demand process is modeled as a Brownian motion. Porteus (1971) studied a discrete-time version of this problem and under the strong condition that the demand distribution belongs to the class of densities that are finite convolutions of uniform and/or exponential densities (note that normal density does not belong to this class), an optimal control policy is a generalized (s, S) policy consisting of a sequence of (si, Si). Using a lower bound approach, we show that an optimal control policy for the Brownian inventory model is determined by a single pair (s, S).
In this paper we analyze a tollbooth tandem queueing problem with an infinite number of servers. A customer starts service immediately upon arrival but cannot leave the system before all customers who arrived before him/her have left, i.e. customers depart the system in the same order as they arrive. Distributions of the total number of customers in the system, the number of departure-delayed customers in the system, and the number of customers in service at time t are obtained in closed form. Distributions of the sojourn times and departure delays of customers are also obtained explicitly. Both transient and steady state solutions are derived first for Poisson arrivals, and then extended to cases with batch Poisson and nonstationary Poisson arrival processes. Finally, we report several stochastic ordering results on how system performance measures are affected by arrival and service processes.
One of the most fundamental results in inventory theory is the optimality of (s, S) policy for inventory systems with setup cost. This result is established under a key assumption of infinite ordering/production capacity. Several studies have shown that, when the ordering/production capacity is finite, the optimal policy for the inventory system with setup cost is very complicated and indeed, only partial characterization for the optimal policy is possible. In this paper, we consider a continuous review production/inventory system with finite capacity and setup cost. The demand follows a Poisson process and a demand that cannot be satisfied upon arrival is backlogged. We show that the optimal control policy has a very simple structure when the holding/shortage cost rate is quasi-convex. We also develop efficient algorithms to compute the optimal control parameters.
There is a growing interest in networks of queues with customers and signals. The signals in these models carry commands to the service nodes and trigger customers to move instantaneously within the network. In this note we consider networks of queues with signals and random triggering times; that is, when a signal arrives at a node, it takes a random amount of time to trigger a customer to move with distribution depending on the source of the signal. By appropriately choosing the triggering times, we can obtain network models such that a signal changes a customer's service time distribution – for example, the signal increases or decreases the service time of a customer. We show that the stationary distribution of this model has a product form solution.
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 ). 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. . 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.
This paper extends product form results for queueing networks with signals to allow history-dependent routing. The signals in these models carry information to nodes and induce multiple customers to move simultaneously within the network. Two models are studied in this paper. In the first one we assume that routing probabilities of a departing customer from a given class of nodes depend on the amount of service just received by the customer and whether its departure is the result of an actual service completion or the result of an arriving signal. In the second model we assume that the routing probabilities of a customer depend on the number of times this customer's service has been interrupted by signals in the past as well as the cause of its departure. We show that both models possess simple product form solutions. These results provide a new dimension in modeling and analyzing practical systems.
For a two-station tandem system with a general arrival process, exponential ervice times, and blocking, we show that the distribution of the departure process does not change when the two stations are interchanged. Blocking here means that for some fixed b≥1, any customer completing service at the first station when b customers are at the second station cannot enter the second queue, and the first station cannot start serving a new customer until a service completion occurs at the second station. This result remains true if arrivals are lost when there are a(a≥0) customers in the system. We also show that when the sum of the service rates is held constant, each departure epoch is stochastically minimized if the two rates are equal. For a and b infinite, our results reduce to those given by Weber  and Lehtonen . Our proof is based on a coupling method first used by Lehtonen. The same results hold for a blocking mechanism in which a customer completing its service at the first station must restart service when b customers are present at the second.
Consider m machines in series with unlimited intermediate buffers and n jobs available at time zero. The processing times of job j on all m machines are equal to a random variable Xj with distribution Fj. Various cost functions are analyzed using stochastic order relationships. First, we focus on minimizing where cj is the weight (holding cost) and Tj the completion time of job j. We establish that if are in a class of distributions we define as SIFR, and and are increasing sequences of likelihood ratio-ordered and stochastic-ordered random variables, respectively, the job sequence [1, 2, … n ] is optimal among all static permutation schedules. Second, for arbitrary processing time distributions, if is an increasing sequence of likelihood ratio-ordered (hazard rate-ordered) random variables and the costs are nonincreasing, then a general cost function is minimized by the job sequence [1,2,…, n] in the stochastic ordering (increasing convex ordering) sense.
Consider a generalized queueing network model that is subject to two types of arrivals. The first type represents the regular customers; the second type represents signals. A signal induces a regular customer already present at a node to leave. Gelenbe  showed that such a network possesses a product form solution when each node consists of a single exponential server. In this paper we study a number of issues concerning this class of networks. First, we explain why such networks have a product form solution. Second, we generalize existing results to include different service disciplines, state-dependent service rates, multiple job classes, and batch servicing. Finally, we establish the relationship between these networks and networks of quasi-reversible queues. We show that the product form solution of the generalized networks is a consequence of a property of the individual nodes viewed in isolation. This property is similar to the quasi-reversibility property of the nodes of a Jackson network: if the arrivals of the regular customers and of the signals at a node in isolation are independent Poisson, the departure processes of the regular customers and the signals are also independent Poisson, and the current state of the system is independent of the past departure processes.
In this article we study the joint optimization of finished goods inventory and pricing in a make-to-stock production system with long-run average profit criterion. The production time is random with controllable rate and the demand is Markovian with rate depending on the sale price. The objective is to dynamically adjust the production rate and the sale price to maximize the long-run average profit. We obtain the optimal dynamic pricing and production control policy and present an efficient bisection algorithm for computing the policy parameters.
Linear birth–death processes with immigration
and emigration are major models in the study of population processes of
biological and ecological systems, and their transient analysis is
important in the understanding of the structural behavior of such
systems. The spectral method has been widely used for solving these
processes; see, for example, Karlin and McGregor . In this article, we provide an
alternative approach: the method of characteristics. This method yields
a Volterra-type integral equation for the chance of extinction and an
explicit formula for the z-transform of the transient
distribution. These results allow us to obtain closed-form solutions
for the transient behavior of several cases that have not been
previously explicitly presented in the literature.
Very few stochastic systems are known to have closed-form
transient solutions. In this article we consider an immigration
birth and death population process with total catastrophes and
study its transient as well as equilibrium behavior. We obtain
closed-form solutions for the equilibrium distribution as well
as the closed-form transient probability distribution at any
time t ≥ 0. Our approach involves solving ordinary
and partial differential equations, and the method of characteristics
is used in solving partial differential equations.
We show that several truncation properties of queueing
systems are consequences of a simple property of censored
stochastic processes. We first consider a discrete-time
stochastic process and show that its censored process has
a truncated stationary distribution. When the stochastic
process has continuous time, we present a similar result
under the additional condition that the process is locally
balanced. We apply these results to single-server batch
arrival batch service queues with finite buffers and queueing
networks with finite buffers and batch movements, and extend
the well-known results on truncation properties of the
and queueing networks with jump-over blocking.
In this paper we extend the notion of quasi-reversibility and apply it to the study of queueing networks with instantaneous movements and signals. The signals treated here are considerably more general than those in the existing literature. The approach not only provides a unified view for queueing networks with tractable stationary distributions, it also enables us to find several new classes of product form queueing networks, including networks with positive and negative signals that instantly add or remove customers from a sequence of nodes, networks with batch arrivals, batch services and assembly-transfer features, and models with concurrent batch additions and batch deletions along a fixed or a random route of the network.
In this paper we consider a network of queues with batch services, customer coalescence and state-dependent signaling. That is, customers are served in batches at each node, and coalesce into a single unit upon service completion. There are signals circulating in the network and, when a signal arrives at a node, a batch of customers is either deleted or triggered to move as a single unit within the network. The transition rates for both customers and signals are quite general and can depend on the state of the whole system. We show that this network possesses a product form solution. The existence of a steady state distribution is also discussed. This result generalizes some recent results of Henderson et al. (1994), as well as those of Chao et al. (1996).
Recently Miyazawa and Taylor (1997) proposed a new class of queueing networks with batch arrival batch service and assemble-transfer features. In such networks customers arrive and are served in batches, and may change size when a batch transfers from one node to another. With the assumption of an additional arrival process at each node when it is empty, they obtain a simple product-form steady-state probability distribution, which is a (stochastic) upper bound for the original network. This paper shows that this class of network possesses a set of non-standard partial balance equations, and it is demonstrated that the condition of the additional arrival process introduced by Miyazawa and Taylor is there precisely to satisfy the partial balance equations, i.e. it is necessary and sufficient not only for having a product form solution, but also for the partial balance equations to hold.
This note introduces reliability issues to the analysis of queueing systems. We consider an M/G/1 queue with Bernoulli vacations and server breakdowns. The server uptimes are assumed to be exponential, and the server repair times are arbitrarily distributed. Using a supplementary variable method we obtain a transient solution for both queueing and reliability measures of interest. These results provide insight into the effect of server breakdowns and repairs on system performance.
Consider a queueing network with batch services at each node. The service time of a batch is exponential and the batch size at each node is arbitrarily distributed. At a service completion the entire batch coalesces into a single unit, and it either leaves the system or goes to another node according to given routing probabilities. When the batch sizes are identical to one, the network reduces to a classical Jackson network. Our main result is that this network possesses a product form solution with a special type of traffic equations which depend on the batch size distribution at each node. The product form solution satisfies a particular type of partial balance equation. The result is further generalized to the non-ergodic case. For this case the bottleneck nodes and the maximal subnetwork that achieves steady state are determined. The existence of a unique solution is shown and stability conditions are established. Our results can be used, for example, in the analysis of production systems with assembly and subassembly processes.
We consider a family of M(t)/M(t)/1/1 loss systems with arrival and service intensities (λt (c), μt (c)) = (λct, μct), where (λt, μt) are governed by an irreducible Markov process with infinitesimal generator Q = (qij)m × m such that (λt, μt) = (λi, μi) when the Markov process is in state i. Based on matrix analysis we show that the blocking probability is decreasing in c in the interval [0, c∗], where c∗ = 1/maxi Σj≠iqij/(λi + μi). Two special cases are studied for which the result can be extended to all c. These results support Ross's conjecture that a more regular arrival (and service) process leads to a smaller blocking probability.
We consider the Klimov model for an open network of two types of jobs. Jobs of type i arrive at station i, have processing times that are exponentially distributed with parameter µi, and when processed either go on to station j with probability pij, or depart the network with probability pi0. Costs are charged at a rate that depends on the number of jobs of the two types in the system. It is shown that for arbitrary arrival processes the policy that gives priority to those jobs for whom the rate of change of the cost function is greatest minimizes the expected cost rate at every time t. This result is stronger than the Klimov result in two ways: arrival processes are arbitrary, and the minimization is at each time t. But the result holds for only two types.