Optimal server scheduling in nonpreemptive finite-population queueing systems

We consider a finite-population queueing system with heterogeneous classes of customers and a single server. For the case of nonpreemptive service, we fully characterize the structure of the server's optimal service policy that minimizes the total average customer waiting costs. We show that the optimal service policy may never serve some classes of customers. For those classes that are served, we show that the optimal service policy is a simple static priority policy. We also derive sufficient conditions that determine the optimal priority sequence.     

A queueing network with a single cyclically roving server

A queueingnetwork that is served by asingle server in a cyclic order is analyzed in this paper. Customers arrive at the queues from outside the network according to independent Poisson processes. Upon completion of his service, a customer mayleave the network, berouted to another queue in the network orrejoin the same queue for another portion of service. The single server moves through the different queues of the network in a cyclic manner. Whenever the server arrives at a queue (polls the queue), he serves the waiting customers in that queue according to some service discipline. Both the gated and the exhaustive disciplines are considered. When moving from one queue to the next queue, the server incurs a switch-over period. This queueing network model has many applications in communication, computer, robotics and manufacturing systems. Examples include token rings, single-processor multi-task systems and others. For this model, we derive the generating function and the expected number of customers present in the network queues at arbitrary epochs, and compute the expected values of the delays observed by the customers. In addition, we derive the expected delay of customers that follow a specific route in the network, and we introduce pseudo-conservation laws for this network of queues.Summary of notation B_i, B _i ^* (s) service time of a customer at queue i and its LST - b_i, b_i ⁽²⁾ mean and second moment of B_i - R_i, R _i ^* (s) duration of switch-over period from queue i and its LST - r_i, r_i mean and second moment of R_i - r, r⁽²⁾ mean and second moment of _i ^N =₁R_i - _i external arrival rate of type-i customers - _i total arrival rate into queue i - _i utilization of queue i; _i=_i - system utilization _i ^N =_1i - c=E[C] the expected cycle length - X _i ^j number of customers in queue j when queue i is polled - X_i=X _i ⁱ number of customers residing in queue i when it is polled - f_i(j) - X _i ^* number of customers residing in queue i at an arbitrary moment - Y_i the duration of a service period of queue i - W_i,T_i the waiting time and sojourn time of an arbitary customer at queue i - F^*(z₁, z₂,..., z_N) GF of number of customers present at the queues at arbitrary moments - F_i(z₁, z₂,..., z_N) GF of number of customers present at the queues at polling instants of queue i - ¯F_i(z₁, z₂,...,z_N) GF of number of customers present at the queues at switching instants of queue i - V_i(z₁, z₂,..., z_N) GF of number of customers present at the queues at service initiation instants at queue i - ¯V_i(z₁,z₂,...,z_N) GF of number of customers present at the queues at service completion instants at queue i The work of this author was supported by the Bernstein Fund for the Promotion of Research and by the Fund for the Promotion of Research at the Technion.Part of this work was done while H. Levy was with AT&T Bell Laboratories.     

Optimal stochastic scheduling in a single server biclass retrial queueing system

The paper deals with the assignment of a single server to two retrial queues. Each customer reapplies for service after an exponentially distributed amount of time. The server operates at customer dependent exponential rates. There are holding costs and costs during service per customer and per unit of time. We provide conditions on which it is optimal to allocate the server to queue 1 or 2 in order to minimize the expected total costs until the system is cleared.     

Optimal server assignment in a two-stage tandem queueing system

We study Markovian queueing systems consisting of two stations in tandem. There is a dedicated server in each station and an additional server that can be assigned to any station. Assuming that linear holding costs are incurred by jobs in the system and two servers can collaborate to work on the same job, we determine structural properties of optimal server assignment policies under the discounted and the average cost criteria.     

Waiting times in queueing networks with a single shared server

We study a queueing network with a single shared server that serves the queues in a cyclic order. External customers arrive at the queues according to independent Poisson processes. After completing service, a customer either leaves the system or is routed to another queue. This model is very generic and finds many applications in computer systems, communication networks, manufacturing systems, and robotics. Special cases of the introduced network include well-known polling models, tandem queues, systems with a waiting room, multi-stage models with parallel queues, and many others. A complicating factor of this model is that the internally rerouted customers do not arrive at the various queues according to a Poisson process, causing standard techniques to find waiting-time distributions to fail. In this paper, we develop a new method to obtain exact expressions for the Laplace–Stieltjes transforms of the steady-state waiting-time distributions. This method can be applied to a wide variety of models which lacked an analysis of the waiting-time distribution until now.     

Optimal control for parallel queues with a single batch server

We consider a queueing system with multiple Poisson arrival queues and a single batch server that has infinite capacity and a fixed service time. The problem is to allocate the server at each moment to minimize the long-run average waiting cost. We propose a Cost-Arrival Weighted (CAW) policy for this problem based on the structure of the optimal policy of a corresponding fluid model. We show that this simple policy enjoys a superior performance by numerical experiments.     

Optimal control of a queueing system with an exponential and an Erlangian server and renewal input stream

Customers arrive in a renewal process at a queue which is served by an exponential and a two-stage Erlangian server. We prove the optimal policy for assignment of customers to the servers which for any t maximizes the expected number of served customers in [0,t].     

Optimal server allocation in general,finite, multi‐server queueing networks

Queueing networks with finite buffers, multiple servers, arbitrary acyclic, series‐parallel topologies, and general service time distributions are considered in this paper. An approach to optimally allocate servers to series, merge, and split topologies and their combinations is demonstrated. The methodology builds on two‐moment approximations to the service time distribution embedded in the generalized expansion method for computing the performance measures in complex finite queueing networks and Powell's algorithm for optimally allocating servers to the network topology. Convexity of the objective function along with results from computational experiments is presented for showing the efficacy of the methodology. Copyright © 2009 John Wiley & Sons, Ltd.     

Stochastic petri net analysis of finite-population vacation queueing systems

We consider queueing systems in which the server occasionally takes a vacation of random duration. The vacation can be used to do additional work; it can also be a rest period. Several models of this problem have been analyzed in the past assuming that the population of the system is infinite. Similarly, it is generally assumed that the capacity of the system is infinite. In this paper we show how the finite-population system can be modeled by the stochastic Petri net. We also extend the model to the finite-capacity system. This research was sponsored by the SDIO Innovative Science and Technology Office and was managed by the Office of Naval Research under grant N3014-88-K-0623.     

Optimal control of single-server queueing networks

A general single-server queueing network model is considered. It is well-known that an optimal policy is determined by the largest-index policy. There is an index for each given queue and one allocates the server to a queue with largest current index. Using discounted dynamic programming we give a new and short proof of this result and derive some characterizations and bounds of the indices. Moreover, it is shown that an approximate largest-index policy yields an approximately optimal policy. These results lead to efficient methods for computing the indices. In particular, we present a general largest-remaining-index method.     

Optimal control of a removable and non-reliable server in an infinite and a finite M/H2/1 queueing system

This paper deals with a single removable and non-reliable server in both an infinite and a finite queueing system with Poisson arrivals and two-type hyper-exponential distribution for the service times. The server may be turned on at arrival epochs or off at service completion epochs. Breakdown and repair times of the server are assumed to follow a negative exponential distribution. Conditions for a stable queueing system, that is steady-state, are provided. Cost models for both system capacities are respectively developed to determine the optimal operating policy numerically at minimum cost. This paper provides the minimum expected cost and the optimal operating policy based on assumed numerical values given to the system parameters, as well as to the cost elements. Sensitivity analysis is also investigated.     

A semidefinite programming approach to the optimal control of a single server queueing system with imposed second moment constraints

Classical analyses of the dynamic control of multi-class queueing systems frequently yield simple priority policies as optimal. However, such policies can often result in excessive queue lengths for the low priority jobs/customers. We propose a stochastic optimisation problem in the context of a two class M/M/1 system which seeks to mitigate this through the imposition of constraints on the second moments of queue lengths. We analyse the performance of two families of parametrised heuristic policies for this problem. To evaluate these policies we develop lower bounds on the optimum cost through the achievable region approach. A numerical study points to the strength of performance of threshold policies and to directions for future research.     

Optimal control of a removable and non-reliable server in an M/M/1 queueing system with exponential startup time

We study a single removable and non-reliable server in the N policy M/M/1 queueing system. The server begins service only when the number of customers in the system reaches N (N1). After each idle period, the startup times of the server follow the negative exponential distribution. While the server is working, it is subject to breakdowns according to a Poisson process. When the server breaks down, it requires repair at a repair facility, where the repair times follow the negative exponential distribution. The steady-state results are derived and it is shown that the probability that the server is busy is equal to the traffic intensity. Cost model is developed to determine the optimal operating N policy at minimum cost.     

Optimal admission control in a queueing system with heterogeneous traffic

We consider optimal admission control of the GI/PH/1-type queueing system. The problem is then reduced to that of determining multi-threshold strategies. Some numerical examples are presented. The results have applications in the optimal input control of information flow in a computer communication network with heterogeneous traffic.     

A queueing system with random server capacity and multiple control

The authors study queueing, input and output processes in a queueing system with bulk service and state dependent service delay. The input flow of customers, modulated by a semi-Markov process, is served by a single server that takes batches of a certain fixed size if available or waits until the queue accumulates enough customers for service. In the latter case, the batch taken for service is of random size dependent on the state of the system, while service duration depends both on the state of the system and on the batch size taken. The authors establish a necessary and sufficient condition for equilibrium of the system and obtain the following results: Explicit formulas for steady state distribution of the queueing process, intensity of the input and output processes, and mean values of idle and busy periods. They employ theory of semi-regenerative processes and illustrate the results by a number of examples. In one of them an optimization problem is discussed.     

A single server queueing system with two phases of service subject to server breakdown and Bernoulli vacation

This paper deals with a single server M/G/1 queue with two phases of heterogeneous service and unreliable server. We assume that customers arrive to the system according to a Poisson process with rate λ. After completion of two successive phases of service the server either goes for a vacation with probability p(0 ? p ? 1) or may continue to serve the next unit, if any, with probability q(=1 ? p). Otherwise it remains in the system until a customer arrives. While the server is working with any phase of service, it may breakdown at any instant and the service channel will fail for a short interval of time. For this model, we first derive the joint distribution of state of the server and queue size, which is one of the chief objectives of the paper. Secondly, we derive the probability generating function of the stationary queue size distribution at a departure epoch. Next, we derive Laplace Stieltjes transform of busy period distribution and waiting time distribution. Finally we obtain some important performance measures and reliability indices of this model.     

Optimal assignment policy of a single server attended by two queues

A single server attends to two separate queues. Each queue has Poisson arrivals and exponential service. There is a switching cost whenever the server switches from one queue to another. The objective is to minimize the discounted or average holding and switching costs over a finite or an infinite horizon. We show numerically that the optimal assignment policy is characterized by a switching curve. We also show that the optimal policy is monotonic in the following senses: If it is optimal to switch from queue one to queue two, then it is optimal to continue serve queue two whenever the number of customers in queue one or in queue two decreases or increases, respectively.