Stable pricing policies in service systems

Summary Customers arriving for service at a service system are allowed to purchase their priorities. A pricing policy is said to be stable if no one customer can reduce his expected cost by deviating from it, provided that all other customers follow it. The policy considered specifies prices to be paid by arrivals based on the number in the system at the epoch of the arrival and a parameter, which is the number of customers in a group paying the same price. The necessary and sufficient conditions for this policy to be stable are derived. The optimal value of the parameter and the prices that will maximize the expected return to the service center operator while maintaining stability are discussed. The basic queueing model assumesPoisson arrivals, exponential service times and single server.

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Zusammenfassung Für die zur Abfertigung an einem Bedienungssystem eintreffenden Kunden besteht die Möglichkeit, ihre gewünschte Abfertigungspriorität gegen einen entsprechenden Preis zu kaufen. Eine vom Betreiber des Bedienungssystems verfolgte Preispolitik wird als stabil bezeichnet, falls kein Kunde seine erwarteten Kosten durch Abweichungen von dieser Preispolitik senken kann, vorausgesetzt daß alle anderen Kunden gemäß dieser Preispolitik behandelt werden. Bei der hier untersuchten Politik werden die von den eintreffenden Kunden zu zahlenden Preise nach der zum Ankunftszeitpunkt im System vorhandenen Kundenzahl festgesetzt. Außerdem wird zusätzlich als Parameter die Anzahl der Kunden in derselben Preisgruppe berücksichtigt. Es werden dann die notwendigen und hinreichenden Bedingungen für die Stabilität der Preispolitik abgeleitet. Anschließend werden diejenigen Werte für den Parameter und die Preise diskutiert, durch die der erwartete Gewinn für den Betreiber des Bedienungssystems bei stabiler Preispolitik maximiert wird. Der Untersuchung wurde ein Warteschlangenmodell mitPoisson-verteilten Ankünften, exponentialverteilten Abfertigungszeiten und einer Bedienungsstation zugrunde gelegt.

     

Sojourn time analysis for a cyclic-service tandem queueing model with general decrementing service

A sojourn time analysis is provided for a cyclic-service tandem queue with general decrementing service which operates as follows: starting once a service of queue 1 in the first stage, a single server continues serving messages in queue 1 until either queue 1 becomes empty, or the number of messages decreases to k less than that found upon the server's last arrival at queue 1, whichever occurs first, where 1 ≤ k ≤ ∞. After service completion in queue 1, the server switches over to queue 2 in the second stage and serves all messages in queue 2 until it becomes empty. It is assumed that an arrival stream is Poissonian, message service times at each stage are generally distributed and switch-over times are zero. This paper analyzes joint queue-length distributions and message sojourn time distributions.     

On a class of systems of quasilinear conservation laws

We consider hyperbolic conservation laws on matrix algebras. We describe entropies of such systems and study properties of generalized entropy solutions and strong generalized entropy solutions to the Cauchy problem. Bibliography: 15 titles. Translated from Problemy Matematicheskogo Analiza, No. 38, December 2008, pp. 73–91.     

Optimal scheduling policies in time sharing service systems

We consider optimal scheduling problems in a TSSS (Time Sharing Service System), i.e., a tandem queueing network consisting of multiple service stations, all of which are served by a single server. In each station, a customer can receive service time up to the prescribed station dependent upper bound, but he must proceed to the next station in order to receive further service. After the total amount of the received services reaches his service requirement, he departs from the network. The optimal policy for this system minimizes the long-run average expected waiting cost per unit of time over the infinite planning horizon. It is first shown that, if the distribution of customer's service requirement is DMRL (Decreasing Mean Residual Life), the policy of giving the highest priority to the customer with the most attained service time is optimal under a set of some appropriate conditions. This implies that any policy without interruptions and preemptions of services is optimal. If the service requirement is DFR (Decreasing Failure Rate), on the other hand, it is shown that the policy of giving the highest priority to the customer with the least attained service time, i.e., the so-called LAST (Least Attained Service Time first) is optimal under another set of some appropriate conditions. These results can be generalized to the case in which there exist multiple classes of customers, but each class satisfies one of the above sets of conditions.     

Conservation laws for a class of third order evolutionary differential systems

Conservation laws of third order quasi-linear scalar evolution equations are studied via exterior differential system and characteristic cohomology. We find a subspace of 2-forms in the infinite prolongation space in which every conservation law has a unique representative. Analysis of the structure of this subspace based upon the symbol of the differential equation leads to a universal integrability condition for an evolution equation to admit any higher order (weight) conservation laws. As an example, we give a complete classification of a class of evolution equations which admit conservation laws of the first three consecutive weights , , . The differential system describing the flow of a curve in the plane by the derivative of its curvature with respect to the arc length is also shown to exhibit the KdV property, i.e., an infinite sequence of conservation laws of distinct weights.

     

Performance of service policies in a specialized service system with parallel servers

This paper considers a scheduling problem occurring in a specialized service system with parallel servers. In the system, customers are divided into the “ordinary” and “special” categories according to their service needs. Ordinary customers can be served by any server, while special customers can be served only by the flexible servers. We assume that the service time for any ordinary customer is the same and all special customers have another common service time. We analyze three classes of service policies used in practice, namely, policies with priority, policies without priority and mixed policies. The worst-case performance ratios are obtained for all of these service policies.     

Polling systems in heavy traffic: Exhaustiveness of service policies

We study the expected delay in cyclic polling models with general ‘branching-type’ service disciplines. For this class of models, which contains models with exhaustive and gated service as special cases, we obtain closed-form expressions for the expected delay under standard heavy-traffic scalings. We identify a single parameter associated with the service discipline at each queue, which we call the ‘exhaustiveness’. We show that the scaled expected delay figures depend on the service policies at the queues only through the exhaustiveness of each of the service disciplines. This implies that the influence of different service disciplines, but with the same exhaustiveness, on the expected delays at the queues becomes the same when the system reaches saturation. This observation leads to a new classification of the service disciplines. In addition, we show monotonicity of the scaled expected delays with respect to the exhaustiveness of the service disciplines. This induces a complete ordering in terms of efficiency of the service disciplines. The results also lead to new rules for optimization of the system performance with respect to the service disciplines at the queues. Further, the exact asymptotic results suggest simple expected waiting-time approximations for polling models in heavy traffic. Numerical experiments show that the accuracy of the approximations is excellent for practical heavy-traffic scenarios. This revised version was published online in June 2006 with corrections to the Cover Date.     

Price setting policies for service systems in case of uncertain demand and service time

Building anM/M/1 queueing model, we determine price and capacity of a service system within monopolistic and competitive frameworks.The existence of a solution for a monopolist is shown and optimal capacity is determined, if demand is elastic.We also determine, within a competitive framework, the number of firms, outputs of each per unit of time, and the relation between price and the number of firms for elastic demands (which has a maximum).Finally, we show that competition supplies a much larger output than monopoly at the price set by the monopolist, if elasticity equals 2.     

The Riemann problem for a class of resonant hyperbolic systems of balance laws

We solve the Riemann problem for a class of resonant hyperbolic systems of balance laws. The systems are not strictly hyperbolic and the solutions take their values in a neighborhood of a state where two characteristic speeds coincide. Our construction generalizes the ones given earlier by Isaacson and Temple for scalar equations and for conservative systems. The class of systems under consideration here includes, in particular, a model from continuum physics that describes the evolution of a fluid flow in a nozzle with discontinuous cross-section.

Résumé

Nous résolvons le problème de Riemann pour une classe de systèmes hyperboliques non-conservatifs et résonants. Ces systèmes ne sont pas strictement hyperboliques et les solutions considérées prennent leurs valeurs au voisinage d'un état constant où deux des vitesses caractéristiques coincident. Notre construction généralise celle donnée précédemment par Isaacson et Temple pour les équations scalaires et les systèmes conservatifs. La classe générale de systèmes étudiée ici comprend, en particulier, un modèle important de la dynamique des milieux continus qui décrit l'évolution d'un fluide dans une tuyère dont la section est discontinue.     

Relative priority policies for minimizing the cost of queueing systems with service discrimination

This paper considers several single-server two-class queueing systems with different cost functions. Customers in the two classes are discriminated by service rates and relative priorities. Most attention is focused on the ones with general quadratic bivariable and exponential cost functions that are usually applied in the relatively complicated systems. To the best of the authors’ knowledge, there is no literature analyzing these two kinds of cost functions on the subject of relative priority. We explicitly present the conditions under which relative priority outperforms absolute priority for reducing system cost and further provide the method to find the optimal DPS policy. Moreover, we also discuss variations where service rates of the two classes are decision variables under service equalization and service discrimination disciplines, respectively.     

Hyperbolic systems of conservation laws with a symmetry

Hyperbolic systems of conservaiton laws with a symmetry are studied. Some peculiar phenomena for such systems are shown. Admissibility criteria for solutions to such systems are discussed. propagation and cancellation of initial oscillations for the systems are classified. As a byproduct of this study, an existence theorem of global solutions for the Cauchy of the systems is established.     

The Riemann problem with delta initial data for a class of coupled hyperbolic systems of conservation laws

In this paper, we solve the Riemann problem with the initial data containing Dirac delta functions for a class of coupled hyperbolic systems of conservation laws. Under suitably generalized Rankine–Hugoniot relation and entropy condition, the existence and uniqueness of solutions involving delta shock waves are proved. Further, four kinds of different structure for solutions are established uniquely.     

On a class of perturbed conservation laws

We investigate behavior of the support and decay for large times of nonnegative entropy solutions of the Cauchy problem: , u=u₀ in{0, ∞}× R Here m > 1, p > 1, and u₀ has compact support. Sharp estimates are given studying the Riemann problem and using comparison results.     

Stationary distributions in a queueing system with vacation times and limited service

This paper deals with a modified M/G/1 queueing system with finite capacity and a walking server. Units waiting are served up to a limited number before the server takes a vacation time and later returns to the queue again. A computational method for the stationary queue length distribution is developed and illustrated with a numerical example. The model was motivated by similar channel access mechanisms in token-ring local area networks.