Workload-Dependent Dynamic Priority for the Multiclass Queue with Reneging

Consider a system with K parallel servers, each with its own waiting room. Upon arrival, a job is routed to the queue of one of the servers. Finding a routing policy that minimizes the total workload in the system is a known difficult problem in general. Even if the optimal policy is identified, the policy would require the full queue length information at the arrival of each job; for example, the join-the-shortest-queue policy (which is known to be optimal for identical servers with exponentially distributed service times) would require comparing the queue lengths of all the servers. In this paper, we consider a balanced routing policy that examines only a subset of c servers, with 1 ≤ c ≤ K: specifically, upon the arrival of a job, choose a subset of c servers with a probability proportional to their service rates, and then route the job to the one with the shortest queue among the c chosen servers. Under such a balanced policy, we derive the diffusion limits of the queue length processes and the workload processes. We note that the diffusion limits are the same for these processes regardless the choice of c, as long as c ≥ 2. We further show that the proposed balanced routing policy for any fixed c ≥ 2 is asymptotically optimal in the sense that it minimizes the workload over all time in the diffusion limit. In addition, the policy helps to distribute work among all the servers evenly.

We consider a priority queue with a dynamic, queue-length-threshold scheduling policy. Customers are classed into two types (type-1 and type-2), and the service order of the two classes depends on the queue length of the type-1 queue. The high priority (type-2) class (e.g., voice) is served until the low priority (e.g., data) queue exceeds the threshold L, at which time service is given to the low priority class until its queue length decreases to L. The arrivals of the two classes follow independent Poisson processes, and the service time of each customer has an exponential distribution with parameter $\mu$. We derive the balance equations in the steady state, and explicitly obtain the joint probability generating function for the queue lengths of the two customer classes. This gives the joint queue length distribution as an integral. We then obtain detailed asymptotic results for the joint distribution. In particular, we study the tail behavior. We also discuss heavy traffic diffusion approximations for this model.

Software rejuvenation is modeled in a client–server system, which provides resources to priority classes of users. To assure availability, resource reservation policies are adopted for the higher priority classes. In addition software rejuvenation is proposed to optimize resource availability. The system is modeled by a cyclic nonhomogeneous Markov chain to capture the variation of the arrival and service rates during a day period. An optimization problem is solved based on a similar previous work and given the optimal resource reservation policy obtained by its solution, rejuvenation is performed and the optimal rejuvenation policy is determined. As a measure of resource availability the blocking probability of each priority class is used. Performability indicators expressing the total cost are also derived, with respect to the optimal resource reservation and optimal rejuvenation policies, to examine whether rejuvenation benefits the system in terms of cost. To derive the blocking probabilities, the limiting probability distribution is computed using explicit generalized approximate inverse preconditioning for solving efficiently sparse linear systems of algebraic equations. Copyright © 2012 John Wiley & Sons, Ltd.

We consider a one-dimensional stochastic control problem that arises from queueing network applications. The state process corresponding to the queue-length process is given by a stochastic differential equation which reflects at the origin. The controller can choose the drift coefficient which represents the service rate and the buffer size b>0. When the queue length reaches b, the new customers are rejected and this incurs a penalty. There are three types of costs involved: A "control cost" related to the dynamically controlled service rate, a "congestion cost" which depends on the queue length and a "rejection penalty" for the rejection of the customers. We consider the problem of minimizing long-term average cost, which is also known as the ergodic cost criterion. We obtain an optimal drift rate (i.e. an optimal service rate) as well as the optimal buffer size b *>0. When the buffer size b>0 is fixed and where there is no congestion cost, this problem is similar to the work in Ata, Harrison and Shepp (Ann. Appl. Probab. 15, 1145---1160, 2005). Our method is quite different from that of (Ata, Harrison and Shepp (Ann. Appl. Probab. 15, 1145---1160, 2005)). To obtain a solution to the corresponding Hamilton---Jacobi---Bellman (HJB) equation, we analyze a family of ordinary differential equations. We make use of some specific characteristics of this family of solutions to obtain the optimal buffer size b *>0.

This paper deals with the analysis of a queueing process involving two classes of customers where the arrival mechanism of each class is subject to a particular control doctrine based on the queue length. Expressions are obtained for the equilibrium distribution of the queue length, the expected queue length and other characteristics which measure the effectiveness of the control doctrine.

This paper concentrates on single server bulk arrival queue system with batch size dependent service and working vacation. The server provides service in two service modes depending upon the queue length. The server provides single service if the queue length is at least ‘a’. On the other hand the server provides fixed batch service if the queue length is at least ‘k’ (k > a). Batch service is provided with some fixed batch size ‘k’. After completion of service if the queue length is less than ‘a’ then the server leaves for working vacation. During working vacation customers are served with lower service rate than the regular service rate. Service during working vacation also contains two service modes. For the proposed model probability generating function of the queue length at an arbitrary time will be obtained by using supplementary variable technique. Some performance measures will also be presented with suitable numerical illustrations.

This paper discusses medium access issues for wireless asynchronous transfer mode (ATM) networks. In particular, a priority scheduling algorithm for a centralized access scheme has been analyzed using the Markov chain theory and the OPNET simulation package. The scheduler allocates resources to the terminals according to their priority class. Different algorithms (first-come first-serve, round robin, select largest queue) are proposed for scheduling the ATM traffic within the same priority class. The performance measures are cell delay, queue length, and cell loss ratio (CLR). It turns out that the scheduler algorithm strongly influences the cell delay which is most important for delay sensitive services. The CLR due to buffer overflows can be kept very low in general except for very bursty sources.

In the modern healthcare system, the rational allocation of emergency department (ED) resources is crucial for enhancing emergency response efficiency, ensuring patient safety, and improving the quality of medical services. This paper focuses on the issue of ED resource allocation and designs a priority sorting system for ED patients. The system classifies patients into two queues: urgent and routine. Considering different service rates, a multi-server preemptive priority queueing model and a multi-server non-preemptive priority queueing model are constructed. Additionally, the number of beds, K, is introduced as the capacity of the urgent queue. By comprehensively considering the costs associated with patient waiting time, the cost of rejecting the most critical patients, and the total costs of beds and servers, a mixed-integer programming model was constructed with the objective of minimizing the total cost. The particle swarm optimization algorithm was applied to determine the optimal number of servers, service rate, and number of beds. Compared with the model proposed by Alipour-Vaezi et al., our model significantly improves patient waiting times and queue lengths using the same data set: the waiting time decreased by 74.44%, by 5.79%, and by 1.13%; the queue length decreased by 78% and by 3.33%. Our model effectively reduces patient waiting times and queue lengths while controlling costs, identifies the optimal number of beds, and achieves optimized resource allocation. Finally, we conducted a sensitivity analysis and provided some valuable management insights.

Non-preemptive priority [formula omitted] queue with servers’ vacations

Many power-aware resource allocation problems in packet networks can be modeled as single-server queueing systems, in which the power consumption depends on the actual service rate. We consider the scenario in which the queue service rate is controlled to minimize server power consumption. We show that power control methods that tune the service rate by using the queue length or the arrival rate exhibit a non-monotonic curve of delay vs. load. This may lead to malfunctioning in end-to-end flow/congestion control protocols, which are based on the assumption that delays increase with increasing load. We propose a new policy, in which the service rate is changed while keeping almost flat the delay curve, which permits to achieve a close-to-optimal trade-off between power and delay.

A significant amount of recent research has been focused on the stability of multiclass open networks of queues (MONQs). It has been shown that these networks may be unstable under various queueing disciplines even when at each one of the nodes the arrival rate is less than the service rate. Clearly, in such cases the expected delay and the expected number of customers in the system are infinite. In this paper we propose a new class of scheduling rules that can be used in multiclass queueing networks. We refer to this class as the stable shortfall-based priority (SSBP) rules. This SSBP class itself belongs to a larger class of rules, which we refer to as the shortfall-based priority (SBP) rules. SBP is a generalization of the standard non-preemptive priority rule in which customers of the same priority class are served first-come, first-served (FCFS). Rules from SBP can mimic FCFS as well as the so-called strict or head-of-the-line priority disciplines. We show that the use of any rule from the SSBP class ensures stability in a broad class of MONQs found in practice. We proceed with the construction of a sample path inequality for the work done by an SSBP server and show how this inequality can be used to derive upper bounds for the delay when service times are bounded. Bounds for the expected delay of each class of customers in an MONQ are then obtained under the assumptions that the external arrival processes have i.i.d. interarrival times, the routings are deterministic and the service times at each step of the route are bounded. In order to derive these bounds for the average delay in an MONQ we make use of some of the classical ideas of Kingman.

Dynamic affinity load balancing of multi-type tasks on multi-skilled servers, when the service rate of each task type on each of the servers is known and can possibly be different from each other, is an open problem for over three decades. The goal is to do task assignment on servers in a real time manner so that the system becomes stable, which means that the queue lengths do not diverge to infinity in steady state (throughput optimality), and the mean task completion time is minimized (delay optimality). The fluid model planning, Max-Weight, and c- $\mu $ -rule algorithms have theoretical guarantees on optimality in some aspects for the affinity problem, but they consider a complicated queueing structure and either require the task arrival rates, the service rates of tasks on servers, or both. In many cases that are discussed in the introduction section, both task arrival rates and service rates of different task types on different servers are unknown. In this work, the Blind GB-PANDAS algorithm is proposed which is completely blind to task arrival rates and service rates. Blind GB-PANDAS uses an exploration-exploitation approach for load balancing. We prove that Blind GB-PANDAS is throughput optimal under arbitrary and unknown distributions for service times of different task types on different servers and unknown task arrival rates. Blind GB-PANDAS desires to route an incoming task to the server with the minimum weighted-workload, but since the service rates are unknown, such routing of incoming tasks is not guaranteed which makes the throughput optimality analysis more complicated than the case where service rates are known. Our extensive experimental results reveal that Blind GB-PANDAS significantly outperforms existing methods in terms of mean task completion time at high loads.

We consider a server serving a time-slotted queued system of multiple packet-based flows, where not more than one flow can be serviced in a single time slot. The flows have exogenous packet arrivals and time-varying service rates. At each time, the server can observe instantaneous service rates for only a subset of flows (selected from a fixed collection of observable subsets) before scheduling a flow in the subset for service. We are interested in queue length aware scheduling to keep the queues short. The limited availability of instantaneous service rate information requires the scheduler to make a careful choice of which subset of service rates to sample. We develop scheduling algorithms that use only partial service rate information from subsets of channels, and that minimize the likelihood of queue overflow in the system. Specifically, we present a new joint subset-sampling and scheduling algorithm called Max-Exp that uses only the current queue lengths to pick a subset of flows, and subsequently schedules a flow using the Exponential rule. When the collection of observable subsets is disjoint, we show that Max-Exp achieves the best exponential decay rate, among all scheduling algorithms that base their decision on the current (or any finite past history of) system state, of the tail of the longest queue. To accomplish this, we employ novel analytical techniques for studying the performance of scheduling algorithms using partial state, which may be of independent interest. These include new sample-path large deviations results for processes obtained by non-random, predictable sampling of sequences of independent and identically distributed random variables. A consequence of these results is that scheduling with partial state information yields a rate function significantly different from scheduling with full channel information. In the special case when the observable subsets are singleton flows, i.e., when there is effectively no a priori channel state information, Max-Exp reduces to simply serving the flow with the longest queue; thus, our results show that to always serve the longest queue in the absence of any channel state information is large deviations optimal.

We study systems with two classes of impatient customers who differ across the classes in their distribution of service times and patience times. The customers are served on a first-come, first-served basis (FCFS) regardless of their class. Such systems are common in customer call centers, which often segment their arrivals into classes of callers whose requests differ in their complexity and criticality. We first consider an M/G/1 + M queue and then analyze the M/M/k + M case. Analyzing these systems using a queue length process proves intractable as it would require us to keep track of the class of each customer at each position in the queue. Consequently, we introduce a virtual waiting time process where the service times of customers who will eventually abandon the system are not considered. We analyze this process to obtain performance measures such as the percentage of customers receiving service in each class, the expected waiting times of customers in each class, and the average number of customers waiting in queue. We use our characterization to perform a numerical analysis of the M/M/k + M system and find several managerial implications of administering a FCFS system with multiple classes of impatient customers. Finally, we compare the performance a system based on data from a call center with the steady-state performance measures of a comparable M/M/k + M system. We find that the performance measures of the M/M/k + M system serve as good approximations of the system based on real data.

The scheduling at an ATM access point is modelled as a stochastic control problem and an optimal scheduling policy is obtained for equalising the input buffer contents in the case of two inputs with unequal arrival rates. The optimal policy is work conserving, and it serves the longer queue when the queue lengths are unequal and serves the queue with the larger arrival rate when the queue lengths are equal. In the case of 0-1 arrival processes, some sample path wise results are obtained.