Some optimality conditions for M b/M/C queueing systems

Heterogeneous traffic types (interactive messages, file transfers, facsimile, etc.) compete for access to a shared resource (satellite channel, bus, etc.). If a packet is not immediately given access to the resource, it waits in a buffer. We assume that the buffer capacity is infinite, that the traffic types arrive according to independent Poisson processes and that the period over which a packet occupies the resource has an arbitrary type-dependent distribution. The traffic types are partitioned into two groups: interactive and noninteractive. The average delay of each of the interactive types is required to below given thresholds. The problem of finding an optimal scheduling policy that minimizes a linear combination of the average delays for the noninteractive types while meeting the design constraints is considered. Simple necessary and sufficient conditions are derived for the existence of a policy that satisfies the constraints. An algorithm is given that decomposes the traffic types into an ordered arrangement of groups, and the existence of a policy that gives strict priority accordingly is established. Under weak conditions on the costs and rates, it is shown that all optimal policies must have this structural property. Sensitivity and aggregation analyses are given. Finally, exploiting the above decomposition, an optimal policy is constructed and is shown to have many appealing properties.

This paper studies the optimal policy for joint control of admission, routing, service, and jockeying in a queueing system consisting of two exponential servers in parallel. Jobs arrive according to a Poisson process. Upon each arrival, an admission/routing decision is made, and the accepted job is routed to one of the two servers with each being associated with a queue. After each service completion, the servers have an option of serving a job from its own queue, serving a jockeying job from another queue, or staying idle. The system performance is inclusive of the revenues from accepted jobs, the costs of holding jobs in queues, the service costs and the job jockeying costs. To maximize the total expected discounted return, we formulate a Markov decision process (MDP) model for this system. The value iteration method is employed to characterize the optimal policy as a hedging point policy. Numerical studies verify the structure of the hedging point policy which is convenient for implementing control actions in practice.

We study the stationary mode of a queuing system (QS) with an infinite storage device, one service device, and exponential service with intensity µ. The input of QS receives a doubly stochastic Poisson flow, the intensity λ(t) of which varies over the interval [α, β] and is a diffusion process with the zero drift coefficient a = 0, diffusion coefficient b > 0, and elastic boundaries α, β. In this paper, two models of a stationary queuing system QS are constructed in the form of systems of differential equations with respect to the characteristics of the applications number. We obtain the necessary and sufficient conditions for the existence and uniqueness of the stationary mode of the QS, the solution of the system of differential equations with respect to the characteristics of the number of applications of the stationary QS, and the non-negativity of the characteristics of the QS. The probability of downtime is estimated. Using the methods of operator analysis, a stationary generating function of the solution in the form of a convergent power series is found.

In this paper, we are interested in solving queueing systems having Poisson batch arrivals, exponential servers and negative customers. Preconditioned Conjugate Gradient (PCG) method is applied to solving the steady-state probability distribution of the queueing system. Preconditioners are constructed by exploiting near-Toeplitz structure of the generator matrix and the Gohberg-Semumcul formula. We proved that the preconditioned system has singular values clustered around one. Therefore Conjugate Gradient (CG) methods when applied to solving the preconditioned system, we expect fast convergence rate. Numerical examples are given to demonstrate our claim.

This dissertation examines an M[x]/G/1 queueing system with a randomized vacation policy and at most J vacations. Whenever the system is empty, the server immediately takes a vacation. If there is at least one customer found waiting in the queue upon returning from a vacation, the server will be immediately activated for service. Otherwise, if no customers are waiting for service at the end of a vacation, the server either remains idle with probability p or leaves for another vacation with probability 1-p. This pattern continues until the number of vacations taken reaches J. If the system is empty by the end of the Jth vacation, the server becomes idle in the system until at least one customer waiting in the queue. In this dissertation, we investigate the following three queueing systems: Reliable server queueing system, un-reliable server queueing system and un-reliable server with a delayed repair queueing system. For the three systems considered in our dissertation, using the supplementary technique, we develop the system size distribution as well as other important system characteristics, such as the system size distribution at busy period initiation epoch, the queue size distribution at a departure epoch, and the distributions of busy period and idle period, etc. Further, for the un-reliable server we also develop main reliability indices of the presented model. Using the renewal reward theorem, a cost model is constructed to determine the optimal randomized vacation policy. Based on the cost model, a heuristic approach is provided to search the joint optimum values of p and J. Some numerical results are presented for illustrative purpose. Our study presents an extension of the existing vacation queueing model and the analysis of the proposed model will provide a useful performance evaluation tool for more general situations arising in real word.

Minimizing Risk Models in Markov Decision Processes with Policies Depending on Target Values

This paper studies the optimal operation of an M/Ek/1 queueing system with a removable service station under steady-state conditions. Analytic closed-form solutions of the controllable M/Ek/1 queueing system are derived. This is a generalization of the controllable M/M/1, the ordinary M/Ek/1, and the ordinary M/M/1 queueing systems in the literature. We prove that the probability that the service station is busy in the steady-state is equal to the traffic intensity. Following the construction of the expected cost function per unit time, we determine the optimal operating policy at minimum cost.

This paper considers a Poisson arrival queuing system with an exponential server (server-1) and a general server (server-2) operating under dual control.We extend our analysis of the same queuing system with an infinite buffer to the finite case.The case presented here has many applications in manufacturing, computer centers and telecommunications systems.

Two different kinds of heavy-traffic limit theorems have been proved for s-server queues. The first kind involves a sequence of queueing systems having a fixed number of servers with an associated sequence of traffic intensities that converges to the critical value of one from below. The second kind, which is often not thought of as heavy traffic, involves a sequence of queueing systems in which the associated sequences of arrival rates and numbers of servers go to infinity while the service time distributions and the traffic intensities remain fixed, with the traffic intensities being less than the critical value of one. In each case the sequence of random variables depicting the steady-state number of customers waiting or being served diverges to infinity but converges to a nondegenerate limit after appropriate normalization. However, in an important respect neither procedure adequately represents a typical queueing system in practice because in the (heavy-traffic) limit an arriving customer is either almost certain to be delayed (first procedure) or almost certain not to be delayed (second procedure). Hence, we consider a sequence of (GI/M/S) systems in which the traffic intensities converge to one from below, the arrival rates and the numbers of servers go to infinity, but the steady-state probabilities that all servers are busy are held fixed. The limits in this case are hybrids of the limits in the other two cases. Numerical comparisons indicate that the resulting approximation is better than the earlier ones for many-server systems operating at typically encountered loads.

In this paper, we consider a queueing system in which there are two exponential servers, each having his own queue, and arriving customers will join the shorter queue. Based on the results given in Flatto and McKean, we rewrite the formula for the probability that there are exactlyk customers in each queue, wherek = 0, 1,…. This enables us to present an algorithm for computing these probabilities and then to find the joint distribution of the queue lengths in the system. A program and numerical examples are given.

In this paper, the waiting time and response time processes in continuous-time two parallel queueing systems are analyzed as well as the system size process in discrete-time two parallel queueing systems. Upon arrival, each job splits into two tasks, denoted by task 1 and task 2. Task k, $k = 1,2$, is assigned to the kth queueing system to be served on a FCFS (first-come-first-served) basis. Each queueing system consists of a single server and an infinite-capacity queue. For the continuous-time queueing system, we assume that jobs arrive in a state-independent Poisson fashion with rate $\lambda $; the servers have exponential service times with rates $\mu _1 $ and $\mu _2 $, respectively. A closed-form expression for the Laplace transform of the ergodic bivariate waiting time distribution is derived. The distribution of the sum of the two waiting times is then obtained. The distribution of the response time when $\mu _1 = \mu _2 $ is derived analytically. For the discrete-time queueing systems, it is assu...

We consider a finite-capacity single-server queuing system in which the problems of capacity design and service quality control are integrated. The amount of time spent on servicing a customer is used as a measure of the quality of service provided. The expected reward from servicing is assumed to be a nondecreasing function of the service duration, while a linear holding cost is assumed for customers waiting in the system. We maximize the long-run average return per unit time by optimally controlling the quality of service to be provided as a function of the workload facing the server for different waiting-room capacities. We show that the optimal control policy for a given capacity is monotone and also examine the effect of varying the system capacity on the optimal control policy. Furthermore, we analyze the design problem of selecting an optimal capacity by assuming that an optimal service control policy will be followed for each system capacity.

In this paper, we analyze a Markovian queueing system with multipletypes of customers and two queues in tandem.All customers have to go through two stages of services.In Stage 1, the queueing system has multiple identical servers while in Stage 2,there is one single-server queue for each type of customers.The queueing discipline in the whole system is Blocked Customer Delayed (BCD).We first obtain the steady-state probability distribution of thequeueing system and the expected waiting time for customers.We then apply the queueing model to solve an optimal pricing policy problemin assuming that the demand rate is dependent on the price.The objective is to minimize the number of serversin the first stage and also maximize the expected earningsby taking into account the demand and the prices.We also obtained some analytic results for the optimal pricing strategy.

After the design and construction of manufacturing tools, adjustment is often necessary to reduce mean deviation from target of product dimensions. We formulate a stochastic dynamic program to find the optimal adjustment policy when the adjustment outcome is uncertain. The optimal policy minimizes the sum of adjustment cost during pre-production and quality cost during production. The formulation is developed for a single process and for two processes producing components for assembly. In the single process case, sufficient conditions are given for the optimal policy to have an upper and lower control limit form. In the two process case, we compare separate and combined policies and give an example where appreciable savings can be realized from a combined policy.

We consider a dynamic pricing model for selling a given stock of a perishable product over a finite time horizon. Customers, whose reservation price distribution changes over time, arrive according to a nonhomogeneous Poisson process. We show that at any given time, the optimal price decreases with inventory. We also identify a sufficient condition under which the optimal price decreases over time for a given inventory level. This sufficient condition requires that the willingness of a customer to pay a premium for the product does not increase over time. In addition to shedding managerial insight, these structural properties enable efficient computation of the optimal policy. Numerical studies are conducted to show the revenue impact of dynamic price policies. Price changes are set to compensate for statistical fluctuations of demand and to respond to shifts of the reservation price. For the former, our examples show that using optimal dynamic optimal policies achieves 2.4–7.3% revenue improvement over the optimal single price policy. For the latter, the revenue increase can be as high as 100%. These results explain why yield management has become so essential to fashion retailing and travel service industries.