Parallel Queues Research Articles

Permuting machines and priority queues

Machines whose sole function is to re-order their input data are considered. Every such machine defines a set of allowable input–output pairs of permutations. These sets are studied in terms of the minimal disallowed pairs (the basis). Some allowable sets with small bases are considered including the one defined by a priority queue machine. For more complex machines defined by two or more priority queues in series or parallel, the basis is proved to be infinite.

Parallel Queue Processor Architecture Based on Produced Order Computation Model

This paper proposes novel produced order parallel queue processor architecture. To store intermediate results, the proposed system uses a first-in-first-out (FIFO) circular queue-registers instead of random access registers. Datum is inserted in the queue-registers in produced order scheme and can be reused. We show that this feature has profound implications in the areas of parallel execution, programs compactness, hardware simplicity and high execution speed. Our performance evaluations show a significant performance improvement (e.g., 10 to 26% decrease in program size and 6 to 46% decrease in execution time over a range of benchmark programs) when compared with the earlier proposed architecture.

Queueing Networks of Random Link Topology: Stationary Dynamics of Maximal Throughput Schedules

In this paper, we study the stationary dynamics of a processing system comprised of several parallel queues and a single server of constant rate. The connectivity of the server to each queue is randomly modulated, taking values 1 (connected) or 0 (severed). At any given time, only the currently connected queues may receive service. A key issue is how to schedule the server on the connected queues in order to maximize the system throughput. We investigate two dynamic schedules, which are shown to stabilize the system under the highest possible traffic load, by scheduling the server on the connected queue of maximum backlog (workload or job number). They are analyzed under stationary ergodic traffic flows and connectivity modulation. The results also extend to the more general case of random server rate.We then investigate the dynamics of acyclic (feed-forward) queueing networks with nodes of the previous type. Their links (connectivities) are stochastically modulated, inducing fluctuating network topologies. We focus on the issue of network throughput and show that it is maximized by simple node server schedules. Rate ergodicity of the traffic flows traversing the network is established, allowing the computation of the maximal throughput.Queueing networks of random topology model several practical systems with unreliable service, including wireless communication networks with extraneous interference, flexible manufacturing systems with failing components, production management under random availability of resources etc.

Calibration of a Queueing Model of RAID Systems

A recent queueing-based modelling methodology of RAID systems compared the mean disk access times of the two most common variants, RAID0-1 and RAID5, as well as a multi-RAID system in which they coexist. Accesses to multiple disks occur concurrently for each logical (user) request and complete only when every disk involved has completed. The models therefore needed to estimate the mean value of the maximum of the individual disk response times, each of which is modelled by the waiting time of an M/G/1 queue. This mean-max value was approximated in terms of the second moment of queueing time which in turn required the third moment of disk service time, itself a function of seek time, rotational latency and block transfer time. To achieve consistently good agreement with an event-driven simulator of the physical hardware and system software requires careful calibration of the resulting model's parameters and validation of its assumptions. This calibration and validation process involves detailed analysis of sub-models to reveal the restrictions necessary on the domain of real-world operating parameters that facilitate a viable predictive model. The process yields significant insight into several of the abstract subsystems involved that may be utilised in a range of practical modelling studies; for example, the effect of approximating a bank of parallel queues with synchronised arrivals by a bank of identical, independent queues. The final comparison against the hardware simulator shows excellent agreement, far surpassing that of the original model.

Convexity in Queues With General Inputs

In this correspondence, we develop fundamental convexity properties of unfinished work and packet waiting time in a queue serving general stochastic traffic. The queue input consists of an uncontrollable background process and a rate-controllable input stream. We show that any moment of unfinished work is a convex function of the controllable input rate. The convexity properties are then extended to address the problem of optimally routing arbitrary input streams over a collection of K queues in parallel with different (possibly time-varying) server rates (/spl mu//sub 1/(t),...,/spl mu//sub K/(t)). Our convexity results hold for stream-based routing (where individual packet streams must be routed to the same queue) as well as for packet-based routing where each packet is routed to a queue by probabilistic splitting. Our analysis uses a novel technique that combines sample path observations with stochastic equivalence relationships.

Structural Flexibility: A New Perspective on the Design of Manufacturing and Service Operations

In this paper, we present a new perspective on flexibility in manufacturing and service operations by exploring a type of operational flexibility that we term “structural flexibility (SF).” We focus on strategic-level issues of how flexibility can be created by using multipurpose resources such as cross-trained labor, flexible machines, or flexible factories. The proposed structural flexibility method uses the structure of the capability pattern to generate indices that quantify the ability of a system to respond to variability in its environment. Simulations of serial and parallel queueing networks provide evidence that this index is useful in predicting the performance rank of alternative designs for implementing multifunctionality in the face of variability. The proposed methodology supports managerial insight into structural design of manufacturing and service systems at the strategic level.

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.

Chained cross-training of workers for robust performance

Training workers to perform multiple tasks can improve workforce agility for dealing with variations in workload. However, cross-training can be costly, time consuming to implement, is limited by worker learning capacity, and can lead to ambiguity about work responsibilities. Therefore, it is important to implement cross-training in the most efficient way and especially, due to the training time required, in a way that is robust to system changes. We use queueing and simulation analysis to investigate cross-training in the context of maintenance in a manufacturing plant. The tasks are independent and can be represented as a set of parallel queues that are served by dedicated and cross-trained workers. We propose a cross-training strategy called chaining, in which a few workers are strategically cross-trained, and show that it yields most of the benefits of cross-training all workers, with much less effort. Most importantly, we demonstrate that cross-training workers to form a “complete chain” is extremely robust in the following ways: (i) it is insensitive to the variety of ways a complete chain can be formed; (ii) it performs well even if there are major changes to or uncertainty in system parameters (such as mean task arrival rates); and (iii) performance is insensitive to control decisions that, without complete chaining, can significantly harm performance.

JOIN MINIMUM COST QUEUE FOR MULTICLASS CUSTOMERS: STABILITY AND PERFORMANCE BOUNDS

We consider a system of K parallel queues providing different grades of service through each of the queues and serving a multiclass customer population. Service differentiation is achieved by specifying different join prices to the queues. Customers of class j define a cost function ψij(ci,xi) for taking service from queue i when the join price for queue i is ci and congestion in queue i is xi and join the queue that minimizes ψij(·,·). Such a queuing system will be called the “join minimum cost queue” (JMCQ) and is a generalization of the join shortest queue (JSQ) system. Non-work-conserving (called Paris Metro pricing system) and work-conserving (called the Tirupati system) versions of the JMCQ are analyzed when the cost to an arrival of joining a queue is a convex combination of the join price for that queue and the expected waiting time in that queue at the arrival epoch. Our main results are for a two-queue system.We obtain stability conditions and performance bounds. To obtain the lower and upper performance bounds, we propose two quasi-birth–death (QBD) processes that are derived from the original systems by suitably truncating the state space. The state space truncation in the non-work-conserving JMCQ follows the method of van Houtum and colleagues. We then show that this method is not applicable to the work-conserving JMCQ and provide sample-path-based proofs to show that the number in each queue is bounded by the number in the corresponding queues of these QBD processes. These sample-path proof techniques might also be of independent interest. We then show that the performance measures like mean queue length and revenue rate of the system are also bounded by the corresponding quantities of these QBD processes. Numerical examples show that these bounds are fairly tight. Finally, we generalize some of these results to systems with more queues.

Optimal routing in two parallel queues

This paper deals with the problem of the computation of the optimal routing sequence in a queuing system made of two parallel queues with exponential service times. The optimal policy (minimizing the expected waiting time) is a Sturmian sequence. An algorithm to compute the optimal routing sequence is proposed together with an extensive set of experiments which show several interesting features of the optimal policy with apparent discontinuities and a fractal behavior. Several good approximations using fast heuristics are given at the end.

Optimal probabilistic routing in distributed parallel queues

We consider the fundamental problem of routing customers among distributed parallel queues to minimize an objective function based on equilibrium sojourn times under general assumptions for the arrival and service processes and under the assumption that customers are routed to the parallel queues in a probabilistic manner. More specifically, we derive explicit solutions for the asymptotically optimal vector of probabilities that control the routing of customers upon arrival among a set of heterogeneous general single-server queues through stochastic-process limits. Our assumption of probabilistic routing is consistent with previous theoretical studies of this optimization problem, and our solutions can be used for the parameter settings of other routing mechanisms found in practice. Stochastic-process limits are exploited in order to be able to handle general arrival and service processes and obtain explicit solutions to the scheduling optimization problems of interest.

A General Decomposition Algorithm for Parallel Queues with Correlated Arrivals

Queueing with correlated arrivals occurs when customers arrive at a set of queues simultaneously. The difficulty in analyzing systems with correlated arrivals is due to the fact that the individual queueing systems are stochastically dependent. Exact methods for analyzing these systems are computationally intensive and are limited to only a few special cases. In this paper, we consider a system of parallel queues with bulk service and correlated arrivals. We show how the matrix-geometric approach can be used to obtain the performance measures of the system. We also develop an algorithm for large systems that efficiently approximates the performance measures by decomposing it into individual queueing systems. Finally, we describe how the principles of our decomposition algorithm can be extended to analyze a variety of different parallel queueing systems with correlated arrivals. We then evaluate the accuracy of our algorithm through a numerical study.

Periodic routing to parallel queues and billiard sequences

In this companion paper of [10] we introduce the combinatorial notion of unbalance for a routing pattern. Using this unbalance we derive an upper bound for the total average expected waiting time of jobs which are routed to parallel queues according to a periodic routing rule. A billiard sequence is obtained with unbalance smaller than or equal to −1, where N is the number of different symbols in the sequence which corresponds to the number of parallel queues in the routing problem.

Queueing and scheduling in random environments

We consider a processing system, composed of several parallel queues and a processor, which operates in a time-varying environment that fluctuates between various states or modes. The service rate at each queue depends on the processor bandwidth allocated to it, as well as the environment mode. Each queue is driven by a job traffic flow, which may also depend on the environment mode. Dynamic processor scheduling policies are investigated for maximizing the system throughput, by adapting to queue backlogs and the environment mode. We show that allocating the processor bandwidth to the queues, so as to maximize the projection of the service rate vector onto a linear function of the workload vector, can keep the system stable under the maximum possible traffic load. The analysis of the system dynamics is first done under very general assumptions, addressing rate stability and flow conservation on individual traffic and environment evolution traces. The connection with stochastic stability is later discussed for stationary and ergodic traffic and environment processes. Various extensions to feed-forward networks of such nodes, the multi-processor case, etc., are also discussed. The approach advances the methodology of trace-based modelling of queueing structures. Applications of the model include bandwidth allocation in wireless channels with fluctuating interference and allocation of switching bandwidth to traffic flows in communication networks with fluctuating congestion levels.

Queueing and scheduling in random environments

We consider a processing system, composed of several parallel queues and a processor, which operates in a time-varying environment that fluctuates between various states or modes. The service rate at each queue depends on the processor bandwidth allocated to it, as well as the environment mode. Each queue is driven by a job traffic flow, which may also depend on the environment mode. Dynamic processor scheduling policies are investigated for maximizing the system throughput, by adapting to queue backlogs and the environment mode. We show that allocating the processor bandwidth to the queues, so as to maximize the projection of the service rate vector onto a linear function of the workload vector, can keep the system stable under the maximum possible traffic load. The analysis of the system dynamics is first done under very general assumptions, addressing rate stability and flow conservation on individual traffic and environment evolution traces. The connection with stochastic stability is later discussed for stationary and ergodic traffic and environment processes. Various extensions to feed-forward networks of such nodes, the multi-processor case, etc., are also discussed. The approach advances the methodology of trace-based modelling of queueing structures. Applications of the model include bandwidth allocation in wireless channels with fluctuating interference and allocation of switching bandwidth to traffic flows in communication networks with fluctuating congestion levels.

The unbalance and bounds on the average waiting time for periodic routing to one queue

In this paper we introduce the combinatorial notion of unbalance for a periodic zero-one splitting sequence. Using this unbalance we derive an upper bound for the average expected waiting time of jobs which are routed to one queue according to a periodic zero-one splitting sequence. In the companion paper [16] the upper bound will be extended to the routing to N parallel queues.

Investigating Effects of Correlated Service Times on System Performance: Implications in Border Crossing Operations

Many transportation systems rely on information technology (IT) for effective and efficient system operations. When the IT systems or resources are shared among various processes or servers, the service times of these processes or servers become correlated. Failure to recognize such correlations in model development may lead to significant inaccuracies. The effects of various types of correlation in the context of border crossing operations were investigated. Several possible reasons for correlation among servers and the ways in which such correlations may affect system operations are explained. Three scenarios are investigated through numerical experimentation: correlated tandem queues, correlated parallel queues, and the impact of correlated parallel queues on a downstream service station. Models are developed with and without consideration for the correlation for these three cases. Ignoring correlation in the performance analysis (i.e., assuming that the servers are independent) leads to inaccurate results, especially when system use is great, that is, at high levels of congestion. In the case of parallel queues, independence assumption leads to overestimated system capacity and to underestimated capacity in tandem queues.

Heavy Traffic Analysis of Two Coupled Processors

We consider two parallel queues. When both are non-empty, they behave as two independent M/M/1 queues. If one queue is empty the server in the other works at a different rate. We consider the heavy traffic limit, where the system is close to instability. We derive and analyze the heavy traffic diffusion approximation for this model. In particular, we obtain simple integral representations for the joint steady state density of the (scaled) queue lengths. Asymptotic and numerical properties of the solution are studied.

A new blocking problem from Java-based schedulers

We consider a form of blocking, which is typical in client–server systems including those implemented under the Enterprise JavaBean (EJB) specification. The novel feature is that tasks must wait for one of a number of parallel queues to clear its outstanding work. Thus, blocking time is the minimum of sojourn times at the parallel queues. Under certain simplifying assumptions, we solve this model for the probability distribution of blocking time and obtain a simple formula for its mean value. We then use this result in an aggregate server model of a larger queueing network in which further non-standard techniques are included to represent this form of blocking. We compare our approximate results against simulation data, obtaining good agreement for both system throughput and queue length probability distributions at equilibrium.

Optimal QoS control of interacting service stations

We consider a system of three queues and two types of packets. Each packet arriving at this system finds in front of it a controller who either sends it in the first queue or rejects it according to a QoS criterion. When the packet finishes its service in the first queue, it is probabilistically routed to one of two other parallel queues. The objective is to minimize a QoS discounted cost over an infinite horizon. The cost function is composed of a waiting cost per packet in each queue and a rejection cost in the first queue. Subsequently, we generalize this problem by considering a system of (m+1) queues and n types of packets. We show that an optimal policy is monotonic.