Fluid analysis of arrival routing

In Hajek's arrival routing problem (1984), customers are routed to one of n queues to minimize average holding cost. Interarrival and service times are exponentially distributed. We solve the associated fluid model. The optimal fluid policy tells us the asymptotic slopes of the switching surfaces in the original problem when the queues are large. If these slopes are nonzero, then numerical tests indicate that the fluid policy performs well in the original stochastic network. The fluid policy also indicates the approximate path that will be taken to recover from large queues: Routing only switches to queues with larger holding cost and once a large queue empties it will remain approximately empty     

Optimal and heuristic policies for dynamic server allocation

We examine a system where the servers in a cluster may be switched dynamically and preemptively from one kind of work to another. The demand consists of M job types joining separate queues, with different arrival and service characteristics, and also different relative importance represented by appropriate holding costs. The switching of a server from queue i to queue j incurs a cost which may be monetary or may involve a period of unavailability. The optimal switching policy is obtained numerically by solving a dynamic programming equation. Two simple heuristic policies—one static and one dynamic—are evaluated by simulation and are compared to the optimal policy. The dynamic heuristic is shown to perform well over a range of parameters, including changes in demand.     

Optimal control of parallel queues with impatient customers

We consider a queueing system with a number of identical exponential servers. Each server has its own queue with unlimited capacity. The service discipline in each queue is first-come-first-served (FCFS). Customers arrive according to a state-dependent Poisson process with an arrival rate which is a non-increasing function of the number of customers in the system. Upon arrival, a customer must join a server’s queue according to a stationary state-dependent policy, where the state is taken to be the number of customers in servers’ queues. No jockeying among queues is allowed. Each arriving customer is limited to a generally distributed patience time after which it must depart the system and is considered lost. Two models of customer behavior are considered: deadlines until the beginning of service and deadlines until the end of service. We seek an optimal policy to assign an arriving customer to a server’s queue. We show that, when the distribution of customer impatience satisfies certain property, the policy of joining shortest queue (SQ) stochastically minimizes the number of lost customers during any finite interval in the long run. This property is shown to always hold for the case of deterministic customer impatience.     

Throughput in Processor-Sharing Queues

Processor-sharing queues are often used to model file transmission in networks. While sojourn time is a common performance metric in the queueing literature, average transmission rate is the more commonly discussed metric in the networking literature. Whereas much is known about sojourn times, there is little known about the average service rate experienced by jobs in processor-sharing queues. We first define the average rate as observed by users and by the queue. In an M/M/1 processor-sharing queue, we give closed-form expressions for these average rates, and prove a strict ordering amongst them. We prove that the queue service rate (in bps) is an increasing function of the minimum required average transmission rate, and give a closed-form expression for the marginal cost associated with such a performance requirement. We then consider the effect of using connection access control by modeling an M/M/1/K processor-sharing queue. We give closed-form expressions for average transmission rates, and discuss the relationship between the queue service rate (in bps), the queue limit, the average rate, and the blocking probability     

Approximate analysis of a discrete-time tandem network of cut-through queues with blocking and bursty traffic

A discrete-time tandem network of cut-through queues is presented. The model allows finite capacity queues, blocking, and bursty traffic. A new bursty arrival process, IBK(k), for cut-through traffic is introduced. The tandem network is analyzed using single-node decomposition. Each queue is analyzed numerically in isolation assuming that its arrival and service processes are known. The parameters of the arrival and service processes of the queues are obtained using an iterative scheme. The results obtained are approximate and validation tests have shown that the model has good accuracy. Using this model, the packet loss, throughput, and queue length distributions were obtained for different traffic parameters and queue sizes.     

Access control to two multiserver loss queues in series

We consider admission policies to two multiserver loss queues in series with two types of traffic. Both are generated according to independent Poisson processes with constant arrival rates. The first type requires service at the first queue and with a positive probability enters the second queue; the second type requires service at only the second queue. The service time distribution is exponential at either station. We show that under appropriate conditions the optimal admission policy that maximizes the expected total discounted reward over an infinite horizon is given by a switching curve. We characterize the form and shape of this curve and its variation with system parameters     

Flow control using the theory of zero sum Markov games

Considers the problem of dynamic flow control of arriving packets into an infinite buffer. The service rate may depend on the state of the system, may change in time, and is unknown to the controller. The goal of the controller is to design an efficient policy which guarantees the best performance under the worst service conditions. The cost is composed of a holding cost, a cost of rejecting customers (packets), and a cost that depends on the quality of the service. The problem is studied in the framework of zero-sum Markov games, and a value iteration algorithm is used to solve it. It is shown that there exists an optimal stationary policy (such that the decisions depend only on the actual number of customers in the queue); it is of a threshold type, and it uses randomization in at most one state     

Joining the right queue: a state-dependent decision rule

The problem of assigning customers to one of several parallel queues so as to minimize the average time spent in the system (sojourn time) is studied as a Markov decision process. It is shown how the approach developed by K.R. Krishman and T.J. Ott (Proc. 25th IEEE Conf. Decision Contr. Dec. 1986, p.2124-8) to investigate state-dependent routing of voice traffic for blocking minimization can also be used for sojourn minimization for data traffic. For queues in parallel, this approach produces a rule, called the `separable' rule, which is a generalization of the `join the shortest queue' rule to the case of dissimilar queues, reducing to the shortest queue rule when the queues are all alike. Numerical results show that in cases where the queues are dissimilar in both the service rates and numbers of their servers, the separable rule is strikingly superior to the shortest queue rule; if the dissimilarities are limited to differences in the service rates, the separable rule practically always is better than the shortest queue rule; if the dissimilarities consist only of the numbers of servers being different, then the shortest queue rule does better than the separable rule in most instances     

Stationary distributions and optimal control of queues with batch Markovian arrival process under multiple adaptive vacations

We consider an infinite-buffer single server queue with batch Markovian arrival process (BMAP) and exhaustive service discipline under multiple adaptive vacation policy. That is, the server serves until system emptied and after that server takes a random maximum number H different vacations until either he finds at least one customer in queue or the server have exhaustively taken all the vacations. The maximum number H of vacations taken by the server is a discrete random variable. We obtain queue-length distributions at various epochs such as, service completion/vacation termination, pre-arrival, arbitrary, post-departure and pre-service. The proposed analysis is based on the use of matrix-analytic procedure to obtain queue-length distribution at a post-departure epoch. Later we use supplementary variable method and simple algebraic manipulations to obtain the queue-length distribution at other epochs using queue-length distribution at post-departure epoch. Some important performance measures, like mean queue lengths and mean waiting times have been obtained. Several other vacation queueing models can be obtained as a special case of our model, e.g., single-, multiple-vacation model and queues with exceptional first vacation time. Finally, the total expected cost function per unit time is considered to determine a locally optimal multiple adaptive vacation policy at a minimum cost.     

Optimal multi-layered congestion based pricing schemes for enhanced QoS

Pricing is an effective tool to control congestion and achieve quality of service (QoS) provisioning for multiple differentiated levels of service. In this paper, we consider the problem of pricing for congestion control in the case of a network of nodes with multiple queues and multiple grades of service.We present a closed-loop multi-layered pricing scheme and propose an algorithm for finding the optimal state dependent price levels for individual queues, at each node. This is different from most adaptive pricing schemes in the literature that do not obtain a closed-loop state dependent pricing policy. The method that we propose finds optimal price levels that are functions of the queue lengths at individual queues. Further, we also propose a variant of the above scheme that assigns prices to incoming packets at each node according to a weighted average queue length at that node. This is done to reduce frequent price variations and is in the spirit of the random early detection (RED) mechanism used in TCP/IP networks.We observe in our numerical results a considerable improvement in performance using both of our schemes over that of a recently proposed related scheme in terms of both throughput and delay performance. In particular, our first scheme exhibits a throughput improvement in the range of 67–82% among all routes over the above scheme.     

An application of modern control theory to a time-dependent queuing system for optimal operation

‘ Bang-bang ’ optimal closed-loop service policy for a time dependent M/M/l queuing system is derived using optimal control theory. The policy is based on probabilistic (and not stochastic) behaviour of the queue. Computational results are obtained for an illustrative example with non-bang-bang service policy using the conjugate gradient algorithm with bounded control variables. It is interesting to note that the optimal service policy is not sensitive to the arrival rate but to the mean service cost of a customer.     

Scheduling with asynchronous service opportunities withapplications to multiple satellite systems

A single server is assigned to M parallel queues with independent Poisson arrivals. Service times are constant, but the server has the opportunity to initiate service at a given queue only at times forming a Poisson process. Four related scheduling policies are investigated: a simple first-come, first-serve policy for which the stability region is determined: a policy with maximum throughput, but requiring the server to have advance knowledge of service opportunities; a policy of threshold type, which is shown to be optimal among nonlookahead policies with preemption; and an adaptive policy, which when M=2 is shown to provide stability for all arrival rate vectors for which stability is possible under any nonlookahead policy with preemption. The work is motivated by the problem of transmission scheduling for a packet-switched, low-altitude, multiple-satellite system     

Dynamic routing of real-time jobs among parallel EDF queues: A performance study

This paper introduces an analytical method for approximating the performance of a firm real-time system consisting of a number of parallel infinite-capacity single-server queues. The service discipline for the individual queues is earliest-deadline-first (EDF). Real-time jobs with exponentially distributed relative deadlines arrive according to a Poisson process. Jobs either all have deadlines until the beginning of service or deadlines until the end of service. Upon arrival, a job joins a queue according to a state-dependent stationary policy, where the state of the system is the number of jobs in each queue. Migration among the queues is not allowed. An important performance measure to consider is the overall loss probability of the system. The system is approximated by a Markovian model in the long run. The resulting model can then be solved analytically using standard Markovian solution techniques. Comparing numerical and simulation results for at least three different stationary policies, we find that the existing errors are relatively small.     

Violation Probability in Processor-Sharing Queues

Processor-sharing queues are often used to model file transmission in networks. While sojourn time is a common performance metric in the queueing literature, average transmission rate is the more commonly discussed metric in the networking literature. Whereas much is known about sojourn times, there is little known about the average service rate experienced by jobs in processor-sharing queues. We focus here upon performance requirements in the form of an upper bound on the probability of failing to achieve a specified minimum transmission rate or a specified minimum average rate. For an M/G/l processor-sharing queue, we give a closed-form expression for this violation probability. We derive closed-form expressions for the marginal service rate with respect to the violation probability and to the minimum transmission rate, and characterize when each is binding. We then consider the effect of using connection access control by modeling an M/G/l/K processor-sharing queue, and discuss the relationship between queue service rate, queue limit, violation probability, and blocking probability. Finally, we consider a two-class discriminatory processor-sharing queue, and discuss what combinations of class weighting and service rate can be used to achieve specified minimum rate violation probabilities for both classes.     

Queueing systems with random service rate

We consider a queue with the arrival process, the service time process and the service rate process as regenerative processes. We provide conditions for its stability, rates of convergence, finiteness of moments and functional limit theorems. This queue can model a queue serving ABR and UBR traffic in an ATM switch; a multiple access channel with TDMA or CDMA protocol and fading; a queue holding best effort or controlled and guaranteed traffic in a router in the integrated service architecture (ISA) of IP-based Internet and a scheduler in the router of a differentiated service architecture. In the process we also provide results for a queue with a leaky bucket controlled bandwidth scheduler. This result is of independent interest. We extend these results to feed-forward networks of queues. We also obtain the results when the arrival rate to the queue can be feedback controlled based on the congestion information in the queue (as in ABR service in the ATM networks or in the real time applications controlled by RTCP protocol in the Internet).     

Dynamic non-preemptive re-allocation policies between two sites with reconfigurable servers

Dynamic server re-allocation can be very useful in real life computing applications. Since the load on many computing systems is not uniformly distributed to each server, it may be effective to transfer the less loaded servers to help the other more loaded ones. However, since transferring takes time, it may not be profitable to actually make the transfer. In this study we model this case with two queues. Each queue is served by one server which can be re-allocated, i.e. an operator may decide to switch a server to serve the other queue. The re-allocation policies we examine are non-preemptive, which implies that a server can be re-allocated if it is idle or has just served a customer. The model is studied with respect to the average cost criterion. We find the optimal re-allocation policy for various instances of the parameters. In addition, we provide a heuristic policy and use simulation experiments to compare it with the optimal one as well as the policy that uses no re-allocation at all.     

Optimal control of the M/G/1 queue with repeated vacations of theserver

An M/G/1 queue where the server may take repeated vacations is considered. Whenever a busy period terminates, the server takes a vacation of random duration. At the end of each vacation, the server may either take a new vacation or resume service; if the queue is found empty, the server always takes a new vacation. The cost structure includes a holding cost per unit of time and per customer in the system and a cost each time the server is turned on. One discounted cost criterion and two average cost criteria are investigated. It is shown that the vacation policy that minimizes the discounted cost criterion over all policies (randomized, history dependent, etc.) converges to a threshold policy as the discount factor goes to zero. This result relies on a nonstandard use of the value iteration algorithm of dynamic programming and is used to prove that both average cost problems are minimized by a threshold policy     

Stability of Parallel Queueing Systems with Coupled Service Rates

This paper considers a parallel system of queues fed by independent arrival streams, where the service rate of each queue depends on the number of customers in all of the queues. Necessary and sufficient conditions for the stability of the system are derived, based on stochastic monotonicity and marginal drift properties of multiclass birth and death processes. These conditions yield a sharp characterization of stability for systems where the service rate of each queue is decreasing in the number of customers in other queues, and has uniform limits as the queue lengths tend to infinity. The results are illustrated with applications where the stability region may be nonconvex.     

Optimal anticipative scheduling with asynchronous transmissionopportunities

Service is provided to a set of parallel queues by a single server. The service of queue i may be initiated only at certain time instances {t_nⁱ}_n=1^∞ that constitute the connectivity instances for queue i. The service of different customers cannot overlap. Scheduling is required to resolve potential contention of services initiated at closely spaced, closer than the service time, connectivity instances. At any time t, the future connectivity instances are available for scheduling. An anticipative policy is given, which at time t schedules the transmissions until a certain future time t+h. The length of the scheduling horizon h is selected based on the backlog at t. The allocation of the server in the interval [t, t+h], is done in accordance to the backlogs of the individual queues at t. The throughput region of the system is characterized, and it is shown that the policy we propose achieves maximum throughput. The policy has a low implementation complexity which is bounded for all the achievable throughput vectors. The average delay and the scheduling complexity are studied by simulation, and the trade-off between the two is demonstrated. The above scheduling problem arises in the access layer of the cross-links of a satellite network     

Optimal routing and buffer allocation for a class of finitecapacity queueing systems

The problem of routing jobs to K parallel queues with identical exponential servers and unequal finite buffer capacities is considered. Routing decisions are taken by a controller which has buffering space available to it and may delay routing of a customer to a queue. Using ideas from weak majorization, it is shown that the shorter nonfull queue delayed (SNQD) policy minimizes both the total number of customers in the system at any time and the number of customers that are rejected by that time. The SNQD policy always delays routing decisions as long as all servers are busy. Only when all the buffers at the controller are occupied is a customer routed to the queue with the shortest queue length that is not at capacity. Moreover, it is shown that, if a fixed number of buffers is to be distributed among the K queues, then the optimal allocation scheme is the one in which the difference between the maximum and minimum queue capacities is minimized, i.e. becomes either 0 or 1