Control and scheduling in a two-station queueing network: Optimal policies and heuristics

Consider a two-station queueing network with two types of jobs: type 1 jobs visit station 1 only, while type 2 jobs visit both stations in sequence. Each station has a single server. Arrival and service processes are modeled as counting processes with controllable stochastic intensities. The problem is to control the arrival and service processes, and in particular to schedule the server in station 1 among the two job types, in order to minimize a discounted cost function over an infinite time horizon. Using a stochastic intensity control approach, we establish the optimality of a specific stationary policy, and show that its value function satisfies certain properties, which lead to a switching-curve structure. We further classify the problem into six parametric cases. Based on the structural properties of the stationary policy, we establish the optimality of some simple priority rules for three of the six cases, and develop heuristic policies for the other three cases.     

On the optimal control of a single-server queueing system: Comment

In a recent paper by Scott and Jefferson, the optimal control of the service rate for a single-server queue with limited waiting space is treated by the maximum principle. We show that their control policies are necessarily suboptimal. Characterizations for optimal control are derived and used to obtain corresponding optimal trajectories in both nonsingular and singular regions.     

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.     

The FCFS service discipline: Stable network topologies, bounds on traffic burstiness and delay, and control by regulators

We consider the well-known First Come First Serve (FCFS) scheduling policy under bursty arrivals. This policy is commonly used in scheduling communication networks and manufacturing systems. Recently it has been shown that the FCFS policy can be unstable for some nonacyclic network topologies.We identify some network topologies under which FCFS is stable for all arrival and service rate vectors within the system's capacity. This is done by determining a sharp bound on the burstiness of traffic exiting from a tandem section of the system, in terms of the burstiness of the incoming traffic. This burstiness bound further allows us to provide a bound on the maximum number of parts in the system, and the maximum delay. It also enables us to analyze the performance of some systems controlled by the use of traffic smoothing regulators. The maximum delay can remain bounded even in the heavy traffic limit, when all stations are 100% utilized.     

Analysis of an MMPP/G/1/K queue with queue length dependent arrival rates,and its application to preventive congestion control in telecommunication networks

Recently telecommunication networks have been designed in order to transfer all types of information services such as voice, data and video. Next generation wireless networks has been developed to integrate the existing technologies and to support comprehensive services. As the traffics of diverse services have properties of timecorrelation and burstiness, unpredictable statistical fluctuation of traffic streams may cause congestion. To suggest a congestion control scheme which controls arrival rates according to the queue length, we consider an MMPP/G/1/K queue with queue length dependent arrival rates. The effect of system parameters on performance measures also is explained with the numerical examples.     

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.     

Convergence of a queueing system in heavy traffic with general patience-time distributions

We analyze a sequence of single-server queueing systems with impatient customers in heavy traffic. Our state process is the offered waiting time, and the customer arrival process has a state dependent intensity. Service times and customer patient-times are independent; i.i.d. with general distributions subject to mild constraints. We establish the heavy traffic approximation for the scaled offered waiting time process and obtain a diffusion process as the heavy traffic limit. The drift coefficient of this limiting diffusion is influenced by the sequence of patience-time distributions in a non-linear fashion. We also establish an asymptotic relationship between the scaled version of offered waiting time and queue-length. As a consequence, we obtain the heavy traffic limit of the scaled queue-length. We introduce an infinite-horizon discounted cost functional whose running cost depends on the offered waiting time and server idle time processes. Under mild assumptions, we show that the expected value of this cost functional for the n-th system converges to that of the limiting diffusion process as n tends to infinity.     

Analysis of a discrete-time queueing system with time-limited service

We analyze a discrete-time, single-server queueing system in which the length of each service period is limited. The server takes a vacation when the limit expires or the queue empties, whichever occurs first. In the former case, the preempted service is resumed after the vacation without loss or creation of any work. This system models the transmission of message frames from a station on timed-token local-area networks (for example, FDDI and IEEE 802.4 token bus). We study the process of the unfinished work and the joint process of the queue size and the remaining service time. By using the technique of discrete Fourier transforms to determine some unknown functions in the governing equations, we numerically obtain exact mean waiting times.A part of the work of H. Takagi was done while he was with IBM Research, Tokyo Research Laboratory.     

An efficient algorithm for optimal design of area traffic control with network flows

An equilibrium network design (EQND) is a problem of finding the optimal design parameters while taking into account the route choice of users. This problem can be formulated as an optimization by taking the user equilibrium traffic assignment as a constraint. In this paper, the methods solving the EQND problem with signal settings are investigated via numerical calculations on two example road networks. An efficient algorithm is proposed in which improvement on a locally optimal search by combining the technique of parallel tangents with the gradient projection method is presented. As it shows, the method combines the locally optimal search and globally search heuristic achieved substantially better performance than did those other approaches.     

Analysis of flexible manufacturing systems with priority scheduling: PMVA

A new methodology for performance analysis of flexible manufacturing systems (FMSs) with priority scheduling is presented. The analytic model developed extends the mean value analysis of closed networks of queues with multiple product types, various non-preemptive priority service disciplines, and with parallel machine stations. Performance measures derived include the expected throughput per product and per station, utilization of machines and transporters, queuing times and queue length measures for various configurations. Extensive numerical calculations have shown that the algorithm used for solving the problem converges rapidly and retains numerical stability for large models. The paper also illustrates the application of the model to a system with a mixture of FCFS and HOL disciplines which gives insights into various priority assignment policies in FMSs. Special attention was given to the problem of scheduling the robot carriers (transporters).     

Optimal control of a queue under a quality-of-service constraint with bounded and unbounded rates

We consider a simple Markovian queue with Poisson arrivals and exponential service times for jobs. The controller chooses state-dependent service rates from an action space. The queue has a finite buffer, and when full, new jobs get rejected. The controller’s objective is to choose optimal service rates that meet a quality-of-service constraint. We solve this problem analytically and compute it numerically under two cases: When the action space is unbounded and when it is bounded.     

Adaptive synchronization to a general non-autonomous chaotic system and its applications

In this paper, new adaptive synchronous criteria for a general class of n-dimensional non-autonomous chaotic systems with linear and nonlinear feedback controllers are derived. By suitable separation between linear and nonlinear terms of the chaotic system, the phenomenon of stable chaotic synchronization can be achieved using an appropriate adaptive controller of feedback signals. This method can also be generalized to a form for chaotic synchronization or hyper-chaotic synchronization. Based on stability theory on non-autonomous chaotic systems, some simple yet less conservative criteria for global asymptotic synchronization of the autonomous and non-autonomous chaotic systems are derived analytically. Furthermore, the results are applied to some typical chaotic systems such as the Duffing oscillators and the unified chaotic systems, and the numerical simulations are given to verify and also visualize the theoretical results.     

Optimal control of generalized multiobjective games with application to traffic networks modeling

The purpose of this paper is to study some new results on the existence and convergence of the solutions to controlled systems of generalized multiobjective games, controlled systems of traffic networks, and optimal control problems (OCPs). First, we introduce the controlled systems of generalized multiobjective games and establish the existence of the solutions for these systems using Browder-type fixed point theorem in the noncompact case and the $C_i$-quasi-concavity. Results on the convergence of controlled systems of the solutions for such problems using the auxiliary solution sets and the extended $C_i$-convexity of the objective functions are studied. Second, we investigate OCPs governed by generalized multiobjective games. The existence and convergence of the solutions to these problems are also obtained. Finally, as a real-world application, we consider the special case of controlled systems of traffic networks. Many examples are given for the illustration of our results.     

Scheduling networks of queues: Heavy traffic analysis of a simple open network

We consider a queueing network with two single-server stations and two types of customers. Customers of type A require service only at station 1 and customers of type B require service first at station 1 and then at station 2. Each server has a different general service time distribution, and each customer type has a different general interarrival time distribution. The problem is to find a dynamic sequencing policy at station 1 that minimizes the long-run average expected number of customers in the system.The scheduling problem is approximated by a dynamic control problem involving Brownian motion. A reformulation of this control problem is solved, and the solution is interpreted in terms of the queueing system in order to obtain an effective sequencing policy. Also, a pathwise lower bound (for any sequencing policy) is obtained for the total number of customers in the network. We show via simulation that the relative difference between the performance of the proposed policy and the pathwise lower bound becomes small as the load on the network is increased toward the heavy traffic limit.     

Sensitivity of optimal prices to system parameters in a steady-state service facility

We consider the problem of maximizing the long-run average reward in a service facility with dynamic pricing. We investigate sensitivity of optimal pricing policies to the parameters of the service facility which is modelled as an M/M/s/K$M/M/s/K$ queueing system. Arrival process to the facility is Poisson with arrival rate a decreasing function of the price currently being charged by the facility. We prove structural results on the optimal pricing policies when the parameters in the facility change. Namely, we show that optimal prices decrease when the capacity of the facility or the number of servers in the facility increase. Under a reasonable assumption, we also show that optimal prices increase as the overall demand for the service provided by the facility increases or when the service rate of the facility decreases. We illustrate how these structural results simplify the required computational effort while finding the optimal policy.     

Application of fuzzy control to a road tunnel ventilation system

This paper deals with the serious problems of ventilation system in a large road tunnel. Higher visibility and lower concentration of carbon monoxide are the key issues concerning the ventilation system. Prior to designing the fuzzy control model, a configuration layout of the ventilation system including sensing, control and traffic prediction as well is conceptually constructed. Based on the layout that offers assignments of sensors and control elements, a fuzzy logic control model is developed. Membership functions of sensor errors and control increments are physically submitted in order to set up the fuzzy logic rules. Timing and spacing filtering in terms of weighting approaches is employed in the fuzzy logic rules. A dynamic equation describing the concentration of air pollution is also given so as to cooperate with the fuzzy logic rules and to play roles in the computer simulation. The result of computer simulation involving five cases indicates that a multi-level scheme is able to solve the engineering problems.     

Analysis of a time-limited service priority queueing system with exponential timer and server vacations

We consider a multi-class priority queueing system with a non-preemptive time-limited service controlled by an exponential timer and multiple (or single) vacations. By reducing the service discipline to the Bernoulli schedule, we obtain an expression for the Laplace-Stieltjes transform (LST) of the waiting time distribution via an iteration procedure, and a recursive scheme to calculate the first two moments. It is noted that we have to select embedded Markov points based on the service beginning epochs instead of the service completion epochs adopted for most of M/G/1 queueing analyses. Through the queue-length analysis, we obtain a decomposition form for the LST of the waiting time in each queue having the exhaustive service.      

On the reliability function of a double redundant system with general repair time distribution

The reliability function is considered for a hot double redundant repairable heterogeneous system with exponentially distributed lifetimes and generally distributed repair times of its components. The problem of its sensitivity to the repair time distribution is discussed.     

On truncations and perturbations of Markov decision problems with an application to queueing network overflow control

Conditions are provided to derive error bounds on the effect of truncations and perturbations in Markov decision problems. Both the average and finite horizon case are studied. As an application, an explicit error bound is obtained for a truncation of a Jacksonian queueing network with overflow control.     

On the departure process of a leaky bucket system with long-range dependent input traffic

Due to the strong experimental evidence that the traffic to be offered to future broadband networks will display long-range dependence, it is important to study the possible implications that such traffic may have for the design and performance of these networks. In particular, an important question is whether the offered traffic preserves its long-range dependent nature after passing through a policing mechanism at the interface of the network. One of the proposed solutions for flow control in the context of the emerging ATM standard is the so-called leaky bucket scheme. In this paper we consider a leaky bucket system with long-range dependent input traffic. We adopt the following popular model for long-range dependent traffic: Time is discrete. At each unit time a random number of sessions is initiated, having the distribution of a Poisson random variable with mean λ. Each of these sessions has a random duration τ, where the integer random variable τ has finite mean, infinite variance, and a regularly varying tail, i.e., P(τ >К) ~ К^-Lα L(К), where 1 < α < 2 L(·) is a slowly varying function. Once a session is initiated, it generates one cell at each unit of time until its termination. We examine the departure process of the leaky bucket policing mechanism driven by such an arrival process, and show that it too is long-range dependent for any token buffer size and any - finite or infinite - cell buffer size. Moreover, upper and lower bounds for the covariance sequence of the output process are established. The above results demonstrate that long-range dependence cannot be removed by the kinds of flow control schemes that are currently being envisioned for broadband networks. This revised version was published online in June 2006 with corrections to the Cover Date.