Empty Queue Research Articles

Stationary analysis of the shortest queue first service policy

We analyze the so-called shortest queue first (SQF) queueing discipline whereby a unique server addresses queues in parallel by serving at any time, the queue with the smallest workload. Considering a stationary system composed of two parallel queues and assuming Poisson arrivals and general service time distributions, we first establish the functional equations satisfied by the Laplace transforms of the workloads in each queue. We further specialize these equations to the so-called “symmetric case,” with same arrival rates and identical exponential service time distributions at each queue; we then obtain a functional equation $$\begin{aligned} M(z) = q(z) \cdot M \circ h(z) + L(z) \end{aligned}$$ for unknown function M, where given functions $$q, L$$ , and $$h$$ are related to one branch of a cubic polynomial equation. We study the analyticity domain of function M and express it by a series expansion involving all iterates of function $$h$$ . This allows us to determine empty queue probabilities along with the tail of the workload distribution in each queue. This tail appears to be identical to that of the head-of-line preemptive priority system, which is the key feature desired for the SQF discipline.

First-Passage Time and Large-Deviation Analysis for Erasure Channels With Memory

This article considers the performance of digital communication systems transmitting messages over finite-state erasure channels with memory. Information bits are protected from channel erasures using error-correcting codes; successful receptions of codewords are acknowledged at the source through instantaneous feedback. The primary focus of this research is on delay-sensitive applications, codes with finite block lengths and, necessarily, non-vanishing probabilities of decoding failure. The contribution of this article is twofold. A methodology to compute the distribution of the time required to empty a buffer is introduced. Based on this distribution, the mean hitting time to an empty queue and delay-violation probabilities for specific thresholds can be computed explicitly. The proposed techniques apply to situations where the transmit buffer contains a predetermined number of information bits at the onset of the data transfer. Furthermore, as additional performance criteria, large deviation principles are obtained for the empirical mean service time and the average packet-transmission time associated with the communication process. This rigorous framework yields a pragmatic methodology to select code rate and block length for the communication unit as functions of the service requirements. Examples motivated by practical systems are provided to further illustrate the applicability of these techniques.

Best Unbiased Estimation and CAN Property in the Stable M/M/1 Queue

The Uniform Minimum Variance Unbiased (UMVU) estimators of ρℓ, the probability of having ℓ or more customers, L, the expected system size, L q , the expected number of customers in the queue, and , the expected number of customers in a non empty queue, are derived based on a random sample of fixed size n on system size at departure points from the geometric distribution on the support {0, 1, 2,…} with mean , which is the distribution of system size in M/M/1 queueing system in equilibrium. The derivations are based on application of Lehmann-Scheffe theorem. Also, CAN estimators of performance measures are derived. In addition the probability distribution of UMVU estimators are obtained.

Cross-Layer Performance Analysis of CSMA/iCA Based Wireless Local Area Network

Carrier sense multiple access with improvised collision avoidance (CSMA/iCA) has recently been proposed as an enhancement to CSMA/CA. It has been reported to be superior than the legacy counterpart in terms of throughput efficiency, packet transmission delay and quantitative fairness index. Nevertheless, the superiority has been shown assuming ideal network conditions: error-free physical layer (L1) and saturated (always non empty) queue at medium access control layer (L2). These strict assumptions, however, do not accurately hold in the real-world Wireless Local Area Networks since the wireless medium is generally error-prone and the arrival of the packets at L2 queue is generally bursty resulting in non-saturated queue occupancy. Thus, the reported performance, especially throughput, in such typical L1/L2 settings is not complete to understand the performance benefit that CSMA/iCA offers under the realistic network settings. In this paper, we relax the aforesaid ideal assumptions and present a cross-layer (L1/L2) performance analysis. Our cross-layer analytical model considers the effect of Rayleigh fading induced bit errors in L1 and non-saturated queue occupancy due to Poisson packet arrival at L2 queue. By virtue of the validated numerical results, we show that the CSMA/iCA consistently retains its throughput gain over CSMA/CA for the non-ideal wireless settings as well.

Asymptotically optimal importance sampling for Jackson networks with a tree topology

This note describes an importance sampling (IS) algorithm to estimate buffer overflows of stable Jackson networks with a tree topology. Three new measures of service capacity and traffic in Jackson networks are introduced and the algorithm is defined in their terms. These measures are effective service rate, effective utilization and effective service-to-arrival ratio of a node. They depend on the nonempty/empty states of the queues of the network. For a node with a nonempty queue, the effective service rate equals the node's nominal service rate. For a node i with an empty queue, it is either a weighted sum of the effective service rates of the nodes receiving traffic directly from node i, or the nominal service rate, whichever smaller. The effective utilization is the ratio of arrival rate to the effective service rate and the effective service-to-arrival ratio is its reciprocal. The rare overflow event of interest is the following: given that initially the network is empty, the system experiences a buffer overflow before returning to the empty state. Two types of buffer structures are considered: (1) a single system-wide buffer shared by all nodes, and (2) each node has its own fixed size buffer. The constructed IS algorithm is asymptotically optimal, i.e., the variance of the associated estimator decays exponentially in the buffer size at the maximum possible rate. This is proved using methods from (Dupuis et al. in Ann. Appl. Probab. 17(4):1306---1346, 2007), which are based on a limit Hamilton---Jacobi---Bellman equation and its boundary conditions and their smooth subsolutions. Numerical examples involving networks with as many as eight nodes are provided.

Corrected asymptotics for a multi-server queue in the Halfin-Whitt regime

To investigate the quality of heavy-traffic approximations for queues with many servers, we consider the steady-state number of waiting customers in an M/D/s queue as s??. In the Halfin-Whitt regime, it is well known that this random variable converges to the supremum of a Gaussian random walk. This paper develops methods that yield more accurate results in terms of series expansions and inequalities for the probability of an empty queue, and the mean and variance of the queue length distribution. This quantifies the relationship between the limiting system and the queue with a small or moderate number of servers. The main idea is to view the M/D/s queue through the prism of the Gaussian random walk: as for the standard Gaussian random walk, we provide scalable series expansions involving terms that include the Riemann zeta function.

Trace-based Derivation of a Lock-Free Queue Algorithm

Lock-free algorithms have been developed to avoid various problems associated with using locks to control access to shared data structures. Instead of preventing interference between processes using mutual exclusion, lock-free algorithms must ensure correct behaviour in the presence of interference. While this avoids the problems with locks, the resulting algorithms are typically more intricate than lock-based algorithms, and allow more complex interactions between processes. The result is that even when the basic idea is easy to understand, the code implementing lock-free algorithms is typically very subtle, hard to understand, and hard to get right.In this paper, we consider the well-known lock-free queue implementation due to Michael and Scott, and show how a slightly simplified version of this algorithm can be derived from an abstract specification via a series of verifiable refinement steps. Reconstructing a design history in this way allows us to examine the kinds of design decisions that underlie the algorithm as describe by Michael and Scott, and to explore the consequences of some alternative design choices.Our derivation is based on a refinement calculus with concurrent composition, combined with a reduction approach, based on that proposed by Lipton, Lamport, Cohen, and others, which we have previously used to derive a scalable stack algorithm. The derivation of Michael and Scott's queue algorithm introduces some additional challenges because it uses a “helper” mechanism which means that part of an enqueue operation can be performed by any process, also in a simulation proof the treatment of dequeue on an empty queue requires the use of backward simulation.

Analysis of two queues in parallel with jockeying and restricted capacities

In this paper, we present two parallel queues with jockeying and restricted capacities. Each exponential server has its own queue, and jockeying among the queues is permitted. The capacity of each queue is restricted to L including the one being served. Customers arrive according to a Poisson process and on arrival; they join the shortest feasible queue. Moreover, if one queue is empty and in the other queue, more than one customer is waiting, then the customer who has to receive after the customer being served in that queue is transferred to the empty queue. This will prevent one server from being idle while the customers are waiting in the other queue. Using the matrix-analytical technique, we derive formulas in matrix form for the steady-state probabilities and formulas for other performance measures. Finally, we compare our new model with some of Markovian queueing systems such as Conolly’s model [B.W. Conolly, The autostrada queueing problems, J. Appl. Prob. 21 (1984) 394–403], M / M / 2 queue and two of independent M / M / 1 queues for the steady state solution.

A Non-Work-Conserving Scheduler to Provide Proportional Delay Differentiated Services and Best Effort Service

The proportional delay differentiation (PDD) model provides consistent packet delay differentiation between classes of service. Currently, the present schedulers performing the PDD model cannot achieve desired delay proportion observed in short timescales under light/moderate load. Thus, we propose a Non-Work-Conserving (NWC) scheduler, which utilizes the pseudo-waiting time for an empty queue and forces each class to compare its priority with those of all other classes. Simulation results reveal that NWC outperforms all current schedulers in achieving the PDD model. However, NWC suspends the server from transmitting packets immediately if an empty class has the maximum priority, resulting in an idle server. Therefore, we further propose two approaches, which will serve a best-effort class during this idle time. Compared with other schedulers, the proposed approaches can provide more predictable and controllable delay proportion, accompanied with satisfactory throughput and average queuing delay.

Flow equivalence and stochastic equivalence in G-networks

G-networks are novel product form queuing networks that, in addition to ordinary customers, contain unusual entities such as “negative customers” which eliminate normal customers, and “triggers” that move other customers from some queue to another. Recently we introduced one more special type of customer, a “reset”, which may be sent out by any server at the end of a service epoch, and that will reset the queue to which it arrives into its steady state when that queue is empty. A reset which arrives to a non-empty queue has no effect at all. The sample paths of a system with resets is significantly different from that of a system without resets, because the arrival of a reset to an empty queue will provoke a finite positive jump in queue length which may be arbitrarily large, while without resets positive jumps are only of size + 1 and they occur only when a positive customer arrives to a queue. In this paper we review this novel model, and then discuss its traffic equations. We introduce the concept of “stationary equivalence” for queueing models, and of “flow equivalence” for distinct queueing models. We show that the flow equivalence of two G-networks implies that they are also stationary equivalent. We then show that the stationary probability distribution of a G-network with resets is identical to that of a G-network without resets whose transition probabilities for positive (ordinary) customers has been increased in a specific manner. Our results show that a G-network with resets has the same form of traffic equations and the same joint stationary probability distribution of queue length as that of a G-network without resets.

Priority queueing model with changeover times and switching threshold

The paper studies a single-server two-queue priority system with changeover times and switching threshold. The server serves queue 1 exhaustively and does not remain at an empty queue if the other one is non-empty. It immediately switches from queue 2 to queue 1 when the length of the latter reaches some level M. Whenever service is changed from one queue to the other a changeover time is required. Arrivals are Poisson, service times and changeover times are independent and exponentially distributed. Using an analytic method we obtain the steady-state joint probability generating function of the lengths of the two queues. By means of this probability generating function some performance measures of the system such as mean length of queue and mean delay can be calculated.

Priority queueing model with changeover times and switching threshold

The paper studies a single-server two-queue priority system with changeover times and switching threshold. The server serves queue 1 exhaustively and does not remain at an empty queue if the other one is non-empty. It immediately switches from queue 2 to queue 1 when the length of the latter reaches some level M. Whenever service is changed from one queue to the other a changeover time is required. Arrivals are Poisson, service times and changeover times are independent and exponentially distributed. Using an analytic method we obtain the steady-state joint probability generating function of the lengths of the two queues. By means of this probability generating function some performance measures of the system such as mean length of queue and mean delay can be calculated.

Delay analysis of a worst-case model of the MetaRing MAC protocol with local fairness

The MetaRing is a medium access control (MAC) protocol for gigabit LANs and MANs with cells removed by the destination stations (slot reuse). Slot reuse increases the aggregate throughput beyond the capacity of single links but may cause starvation. In order to prevent this the MetaRing MAC protocol includes a fairness mechanism. Two types of fairness algorithms have been proposed: `global' and `local'. The MetaRing analysed in this paper implements the local fairness algorithm specified in [2]. In order to reduce the complexity we have identified a simplified model which can be analytically solved and yet still provides useful information on network performances. This model represents a worst-case scenario in which network congestion is stressed, i.e. no station, apart from a specific station (tagged station), ever has an empty queue. The model proposed can be represented by a discrete time discrete state Markov chain of M/G/1-type and hence the matrix analytical methodology has been used to solve it. Our analysis focuses on the average access delay experienced by the tagged station as a function of the offered load. The results show that average access delay remains bounded for values of offered load less than or equal to 90%. Furthermore, the average access delay depends on the number of interfering stations and on the protocol parameter Q r.

Sticky States in Banyan Network Queues and Their Application to Analysis

A finite-buffered banyan network analysis technique designed to model networks at high traffic loads is presented. The technique specially models two states of the network queues: queue empty and queue congested (roughly, zero or one slots free). A congested queue tends to stay congested because packets bound for the queue accumulate in the previous stage. The expected duration of this state is computed using a probabilistic model of a switching module feeding the congested queue. A technique for finding a lower arrival rate to an empty queue is also described. The queues themselves are modeled using independent Markov chains with an additional congested state added. The new analysis technique is novel in its modeling the higher arrival rate to a congested queue and a lower arrival rate to an empty queue. Comparison of queue state distributions obtained with the analysis and simulation shows that an important feature of congestion is modeled.

Novel architecture for ATM QoS management

A novel architecture and an enhanced approach to a flexible, starvation-free ATM QoS managing function is proposed. To meet the stringent timing constraint of the delay QoSs, both high-speed sorter and subtractor are exploited to sort the priority value as well as to process the priority value. The subtractive policy is used to prevent starvation and to assign the priority of each output queue. In addition, to prevent underflow of priority value and to process the empty queue situation, both renormalisation circuit and empty flag are exploited to perform the normalisation function. Simulation results show that the throughput of the enhanced architecture is more than 50 M output requests per second (21.2 Gbit/s for ATM cells) by using a 1.2 µm CMOS process. The proposed architecture can be cascadable, making it very suitable for complex QoS management.

Geometric equilibrium distributions for queues with interactive batch departures

Gelenbe et al. [1, 2] consider single server Jackson networks of queues which contain both positive and negative customers. A negative customer arriving to a nonempty queue causes the number of customers in that queue to decrease by one, and has no effect on an empty queue, whereas a positive customer arriving at a queue will always increase the queue length by one. Gelenbe et al. show that a geometric product form equilibrium distribution prevails for this network. Applications for these types of networks can be found in systems incorporating resource allocations and in the modelling of decision making algorithms, neural networks and communications protocols.

Markovian parallel system with the repairman's vacation

A parallel Markovian model with R repairmen and a vacation for the repairmen is considered. In this paper two models are analysed. In the first model, a repairman takes only a single vacation at a time. In the second model a repairman returning to an empty queue immediately takes another vacation. A recursive computational approach is given to solve the steady state equations of both models. Numerical results for performance measures are presented.

Implementing queues in Lisp

A queue is a data structure where items are entered one at a time and removed one at a time in the same order---i.e., first in first out. They are the same as stacks except that in a stack, items are removed in the reverse of the order they are entered---i.e., last in first out. Queues are most precisely described by the functions that act on them: (make-queue) Creates and returns a new empty queue.

Analyses of an M/M/N/ queue with server's vacations

Consider an M/M/N queueing system. A server that completes service and finds no waiting customers leaves for a vacation whose length follows an exponential distribution. A server returning to an empty queue takes another vacation. In this paper, we present a matrix-geometric approach for modelling the system and propose algorithms for computing the stationary queue length distribution, the expected length of busy period, the mean waiting time, and the waiting time distribution. Two numerical examples are given at the end.

Is Multiple Sclerosis an Infectious Disease? Inference about an Input Process Based on the Output

The output process of an infinite-server queue with a Poisson process input is observed starting at time 0 with an empty queue. It is assumed that the service time distribution is known. This article discusses statistical inference about the input intensity. A controversial issue in the study of multiple sclerosis is addressed as a motivation for the model and methods developed.