Ergodic theorems for queuing systems with dependent inter-arrival times

We study a G/GI/1 single-server queuing model with i.i.d. service times that are independent of a stationary process of inter-arrival times. We show that the distribution of the waiting time converges to a stationary law as time tends to infinity provided that inter-arrival times satisfy a Gärtner-Ellis type condition. A convergence rate is given and a law of large numbers established. These results provide tools for the statistical analysis of such systems, transcending the standard case with independent inter-arrival times.     

Optimal arrangements of queuing systems with FIFO service discipline

The paper studies a class of queuing systems with calls occurring in a certain space. Most frequently, a flow of homogeneous requests that differ only in the moments of their arrival at the system are considered. A specific feature of the class of systems under consideration is the necessity of using the information on the positions of serving devices as well as on the positions of the entering calls and their density distribution. Such models of the queuing systems are used for studying real systems in which the service is made by the objects placed over a territory.     

Regions of stability of queuing systems

A method of stability analysis is considered for the solution of the inverse queuing problem, and quantitative estimates are obtained.Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei — Trudy Seminara, pp. 107–115, 1980.The author expresses his gratitude to V. V. Kalashnikov for a useful discussion of this paper.     

On queuing systems with failures and several types of input flows

In this paper two models of queuing systems with failures and several types of input flows are considered. These models can simulate the functioning of cellular communications networks under various control regimes. For stationary probabilities of service process (including failed ones) the closed form and the effective calculating algorithms are obtained. The results are used to control the input flows in an optimal way. This research was supported by the INTAS (grant 96-0828).     

Stationary distribution for repairable systems

Stationary distribution of repairable systems with limited service and spare devices is considered on an equilibrium set by using the Gibbs potential representation. As an application we consider the reliability of k‐out‐of‐n system. Copyright © 2000 John Wiley & Sons, Ltd.     

A central limit theorem for mabkovian queuing systems

In the paper a central limit theorem for the total busy time of a Markovian queuing system is proved (provided that the system satisfies some simple assumptions), This is based on a stochastic convergence theorem for the average number of services. The latter is presented in a form of “law of large numbers” as well. The case of bounded queue length is particularly analysed.     

A two-population game in observable double-ended queuing systems

In this study, the existing game theoretical framework is extended to strategic queuing in search of solutions for a two-population game in observable double-ended queuing systems with zero matching times. We show that multiple Nash equilibria and one unique subgame perfect Nash equilibrium exist in this game.     

On the discounted cost of Markovian queuing systems

The paper considers Markovian queues. A rarely discussed characteristic, the discounted cost is analysed. Some continuity properties of the distribution function of he discounted cost are proved. The method used here seems to be applicable to more indicated systems as well.     

Numerical performance of approximate queuing formulae with application to flexible manufacturing systems

This paper calls attention to two of the more successful queuing approximation formulae — one by Kramer and one by Marchal. The analytic solution of a range of single server Erlang cases is compared to the two approximation formulae. Then a family of H₂/M/1 cases is similarly considered. Maximum errors are seen to be about three percent. The Kramer formula seems to be better when the interarrival coefficient of variation is less than 0.66 and the Marchal formula is better for larger interarrival coefficients of variation. Finally, a multiserver refinement function (the ratio of G/G/1 results to M/M/1 results) is proposed to scale M/M/s as an approximation for G/G/s. In most of these multiple channel cases, the maximum error is less than six percent. The last section of this paper presents a simple, representative FMS. It is modelled as an open queuing network. Then the approximation procedure is applied node by node to illustrate the estimation of system performance measures such as machine utilizations and throughput.     

Compound transient phenomena in multiphase queuing systems. II

The paper continues the article by the author on functional limit theorems in queuing systems under heavy traffic. Theorems are proved for the virtual process of serving jobs when at phases of the queuing system various conditions of heavy traffic are satisfied (compound transient phenomena). Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 3, pp. 343–356, July–September, 1999. Translated by Z. Kryžius     

A closed queuing maintenance network for automated manufacturing systems

This paper models an unreliable automated manufacturing system (AMS) by a closed queuing network. The AMS consists of a multi-stage network of automated work stations linked by a computer. A closed queuing algorithm is applied to determine the system availability under steady state for the AMS. This algorithm is then integrated into a cost optimization model. By applying the revised genetic algorithm, the optimal (or near-optimal) number of standby units and repair rates for the repair stations are derived by minimizing the total cost. The model is verified by the intuitive results from the sensitivity analysis. A numerical example is used to compare the revised genetic algorithm and the conventional genetic algorithm. The results show that the proposed revised algorithm leads to significant improvement in execution time and lower average total cost.     

Stationary states of random Hamiltonian systems

Summary We investigate the ergodic properties of Hamiltonian systems subjected to local random, energy conserving perturbations. We prove for some cases, e.g. anharmonic crystals with random nearest neighbor exchanges (or independent random reflections) of velocities, that all translation invariant stationary states with finite entropy per unit volume are microcanonical Gibbs states. The results can be utilized in proving hydrodynamic behavior of such systems.Hill Center for Mathematical Sciences, Rutgers University, New Brunswick, NJ 08903, USAJF was supported in parts by Japan Society for Promotion of Science (JSPS) and by NSF Grant DMR89-18903     

Parallel queuing systems with and without decentralized server reallocation — A simulation study

This paper describes the results of an initial exploratory research effort on the subject of the dynamic reallocation of servers in a group of parallel queuing systems under decentralized control. Each queuing system is described as being (M/E_k/c_i,t): (FCFS/∞/∞) where t = 1,.., ∞ and i = 1,..., I. Server reallocation is handled by a (G^(b)/G/C): (FCS/C/K) system — where C = Σ_t=1^∞Σ_i=1^Ic_i,t and K is an unknown constant — located at a higher level than the group members. The customers of the latter system are requests for servers independently generated by identical heuristic event-triggered operating policies at each of the lower level systems. These local policies are of the stochastic review, relative control number type. Performance is measured by means of a variety of criteria from the viewpoint of the group as a whole (global) as well as an individual queuing system (local). Results of experiments using a simulation model confirmed the null hypothesis that the customer service behaviour of a group that shared servers was significantly different from the behaviour of a control group that did not share servers.     

Stochastically minimizing the number of customers in exponential queuing systems

This paper deals with the problem of controlling an exponential queuing system (that is, a system with exponential service times and Poisson arrivals) so as to stochastically minimize the number of customers in the system at any time t > 0. Sufficient (simple) conditions are developed for a policy to be optimal. Similar conditions are sufficient for a policy to stochastically minimize (maximize) any function of the state of the system. Two models are considered to illustrate the results. In both cases, optimal policies are shown to satisfy these conditions by a simple inductive procedure.