Random Queue Research Articles

Impact of TCP-like congestion control on the throughput of multicast groups

We study the impact of random queueing delays stemming from traffic variability on the performance of a multicast session. With a simple analytical model, we analyze the throughput degradation within a multicast (one-to-many) tree under TCP-like congestion and flow control. We use the (max,plus) formalism together with methods based on stochastic comparison (association and convex ordering) and on the theory of extremes to prove various properties of the throughput. We first prove that the throughput predicted by a deterministic model is systematically optimistic. In the presence of light-tailed random delays, we show that the throughput decreases according to the inverse of the logarithm of the number of receivers. We find analytically an upper and a lower bound for the throughput degradation. Within these bounds, we characterize the degradation which is obtained for various tree topologies. In particular, we observe that a class of trees commonly found in IP multicast sessions is significantly more sensitive to traffic variability than other topologies.

ON PARALLEL QUEUING WITH RANDOM SERVER CONNECTIVITY AND ROUTING CONSTRAINTS

We study systems of parallel queues with finite buffers, a single server with random connectivity to each queue, and arriving job flows with random or class-dependent accessibility to the queues. Only currently connected queues may receive (preemptive) service at any given time, whereas an arriving job can only join one of its accessible queues. Using the coupling method, we study three key models, progressively building from simpler to more complicated structures.In the first model, there are only random server connectivities. It is shown that allocating the server to the Connected queue with the Fewest Empty Spaces (C-FES) stochastically minimizes the number of lost jobs due to buffer overflows, under conditions of independence and symmetry.In the second model, we additionally consider random accessibility of queues by arriving jobs. It is shown that allocating the server to the C-FES and routing each arriving job to the currently Accessible queue with the Most Empty Spaces (C-FES/A-MES) minimizes the loss flow stochastically, under similar assumptions.In the third model (addressing a target application), we consider multiple classes of arriving job flows, each allowed access to a deterministic subset of the queues. Under analogous assumptions, it is again shown that the C-FES/A-MES policy minimizes the loss flow stochastically.The random connectivity/accessibility aspect enhances significantly the structure and application scope of the classical parallel queuing model. On the other hand, it introduces essential additional dynamics and considerable complications. It is interesting that a simple policy like FES/MES, known to be optimal for the classical model, extends to the C-FES/A-MES in our case.

Random Queues in Signalized Road Networks

Average delay is an important criterion of evaluating traffic signals. Traffic signals, stream merges, and stream splits influence vehicular delay in signalized networks through influencing random fluctuation of traffic. This paper proposes a method of estimating the random component of queues and delays at signalized intersections inside a network of urban streets. Equations describing the effect of signals and traffic merging and splitting are derived, as well as a comprehensive equation combining the considered effects. Fixed speeds and capacities are assumed. The derived comprehensive equation is used to discuss certain cases of signalized streets. The equation is evaluated using the Monte Carlo technique. The proposed method can be incorporated into existing procedures for evaluating and optimizing traffic signals.

Sojourn times in a random queue with and without preemption

We consider an equilibrium M/M/1 queue in which customers arrive at a random position, both for the preemptive and non-preemptive cases. The Laplace–Stieltjes transforms of their sojourn time distributions are derived and in the latter case it is verified that the result is that of an M/M/1 queue where the server chooses the next task to serve at random.

Equilibrium Strategies for Processor Sharing and Random Queues with Relative Priorities

We consider a memoryless single-server queue in which users can purchase relative priority so as to reduce their expected waiting costs, which are linear with time. Relative priority is given in proportion to a price paid by customers present in the system. For two service disciplines, (weighted) processor sharing and (weighted) random entrance, we find the unique pure and symmetric Nash equilibrium price paid by the customers.

Overflow Delay Model for Signalized Arterials

Delay is the principal parameter used to measure the performance of signalized intersections. Average stopped delay is the primary measure of effectiveness used in evaluating level of service at signalized intersections in the U.S. Highway Capacity Manual (HCM). In an urban street network, the traffic performance at contiguous intersections is highly correlated. Typically, two upstream effects are present, signal coordination and signal metering. Therefore, the delay models developed for an isolated intersection with the assumption of random arrivals are no longer appropriate for delay estimation at a signalized arterial. The impact of signal coordination on delay has been well documented in the literature. However, the signal metering effect has yet to be thoroughly evaluated. For uncongested network flow conditions, the upstream metering effect is of less importance. However, for congested network flow conditions operating at or slightly below capacity, upstream metering becomes critical. In this study, an analytical overflow delay model that effectively incorporates the upstream metering effect for congested signalized networks is developed and the parameters are calibrated. The model is founded on a random queue model that assumes a generalized arrival distribution and a bulk service time distribution. The format of the overflow delay model developed in this work is consistent with the format of the current HCM model. The enhanced model, when effectively validated by field data, can be a good supplement for the current HCM model.

Transmission of compressed voice over integrated services frame relay networks: priority service and adaptive buildout delay

Voice packets must be delivered within a time constraint to be played out at the receiver, and the simplicity of frame relay (FR) protocols makes this difficult to achieve with a low loss rate for integrated traffic. Highly compressed voice is more sensitive to losses than uncompressed voice and also changes the nature of the traffic in a FR network because of the short length of the voice frames. Priority for voice is shown to be highly desirable to achieve high voice performance where low-speed access links may be used for integrated service traffic. Packetised voice transmission also requires that the receiver compensate for the random queuing delays experienced by each voice packet. The packetised voice protocol (PVP) achieves this by using delay stamps created in packet switching nodes at the packet level, but this is not possible in FR networks because they have no packet level. The authors propose an adaptive buildout delay mechanism which requires no action by the switching nodes, meets the requirements of random delay compensation with acceptable loss rates, and achieves shorter end-to-end voice delay than the PVP.

NASH EQUILIBRIUM AND BUYER RATIONING RULES: EXPERIMENTAL EVIDENCE

Laboratory experiments compare the results of two different buyer rationing rules on Bertrand‐Edgeworth duopoly markets. The two rationing rules, a value queue and a random queue, specify the ordering of buyers and generate different predictions under otherwise identical conditions. These experiments show market outcome can chane significantly with a change in the ordering of buyers. The direction of these changes agrees with the predictions of the stage‐game Nash equilibrium. Thus, the single‐shot Nash equilibrium is show to have predictive power even in repeated games.

Reneging from Processor Sharing Systems and Random Queues

In a memoryless processor sharing system and in a memoryless random queue, both with some cost structure, it is shown that there exists an ε-Nash equilibrium symmetric reneging strategy which is Markovian and stationary with respect to the queue length and which is of the following (N*, θ*, η(ε)) form: (i) stay in the system when the queue size is smaller than some specified critical level N*; (ii) randomly renege with constant specified rate θ* when the queue size is N*; (iii) randomly renege with large enough rate ηn(ε) when the queue size is n with n>N*. Moreover, for all n with n>N*, limε→ 0ηn(ε)=∞. On the other hand, N* and θ* are not functions of ε and they depend only on the parameters defining the system. Finally, a stationary and Markovian symmetric Nash equilibrium strategy does not exist.

Reversibility and stochastic networks

This paper considers the M/M/k queueing system where a line is being formed in front of each server. Arriving customers join the shorter line and switch to another once the latter becomes shorter. Since the expected waiting time equals that in the first come first served case (FCFS), it is of interest to compute the variance of the waiting time. We compute it explicitly for the cases where k = 1, k = 2, k = 3 and for the case of a random queue. For the case k ≥ 4, we express the variance in terms of the roots of a k-degree polynomial.

The spitzer identity for some bivariate processes in queueing theory: an algebraic approach

This note deals with a purely algebraic development of the Spitzer identity for a bivariate Markov process , which is important for random walks, dams and queues. This identity, comprising the results of Kingman on the process , has been known for a long time in an analytic form as a solution of a certain integral equation. The algebraic approach leads to explicit results of interesting structure. In particular, the bivariate distribution of (Yn, Zn) can be expressed in terms of the marginal distributions of the stochastically dependent variables Yn and Zn.

Pasion

PASION is a process and event-oriented simulation language designed for those who already know and use Pascal. The lan guage has a two-level (process/event) structure and permits all the Pascal structures. PASION provides all necessary facilities needed to handle sequences of random events, queues, and quasi-parallel processes, both discrete and continuous. A PASION program consists of a sequence of process declara tions. At run-time the program generates objects that represent model processes due to the process declarations. Compared with Simula, PASION is rather simple and offers similar compromise between simplicity and flexibility as does Pascal when compared to Algol.

Suggestions for control mechanism for reducing the number of priority changes in an adaptive loop

A modified Pierce loop network has been described in an earlier paper, incorporating adaptive flow control. This involves a full duplex loop with two types of time slots, one for handling data packets and the other for the transmission of control packets. Each node adjusts its slot usage priority according to its current queue size. A loop supervisor maintains a record of node priorities, and assigns appropriate priority to slots. Since each priority change creates a control message, the number of control packets can become unnecessarily large and eventually cause congestion. In the paper, we describe the use of a statistical mechanism for reducing the number of unnecessary priority changes but with little adverse effect on data-traffic performance. The effectiveness of such a mechanism is evaluated in terms of its ability to prevent overreaction to random queue fluctuation while reacting correctly when a priority change is warranted.

A Queueing Model for Traffic Flow

Summary A model for queue formation and disintegration is proposed for traffic on a road where flow is unimpeded. The model assumes that vehicles are travelling in random queues and a distribution for queue length is derived having as a single parameter the ratio of the rate of mergings between pairs of queues and the rate of overtaking from queues of more than one vehicle. The first moment and maximum likelihood estimator associated with the distribution are derived and compared in a simulation study. The distribution is then fitted to data collected in different sites, with varying results.

Discrete queues with one server

In this paper mathematical methods are given for finding the distributions of the virtual waiting time and the occupation time for single-server queues in which the interarrival times and the service times are discrete random variables taking on positive or non-negative integral values. First we shall consider deterministic queues in which the interarrival times and the service times are numerical (non-random) quantities and deduce a fundamental relation for such queues. Then we shall consider random queues with recurrent input and general service times and study the time dependent behavior of such queues.

Discrete queues with one server

In this paper mathematical methods are given for finding the distributions of the virtual waiting time and the occupation time for single-server queues in which the interarrival times and the service times are discrete random variables taking on positive or non-negative integral values. First we shall consider deterministic queues in which the interarrival times and the service times are numerical (non-random) quantities and deduce a fundamental relation for such queues. Then we shall consider random queues with recurrent input and general service times and study the time dependent behavior of such queues.

A relation between stationary queue and waiting time distributions

A theorem is proved which, in essence, says the following. If, for any queueing system, (i) the arrival process is stationary, (ii) the queue discipline is first-in-first-out (FIFO), and (iii) the waiting time of each customer is statistically independent of the number of arrivals during any time interval after his arrival, then the stationary random queue size has the same distribution as the number of customers who arrive during a random time interval distributed as the stationary waiting time.

A relation between stationary queue and waiting time distributions

A theorem is proved which, in essence, says the following. If, for any queueing system, (i) the arrival process is stationary, (ii) the queue discipline is first-in-first-out (FIFO), and (iii) the waiting time of each customer is statistically independent of the number of arrivals during any time interval after his arrival, then the stationary random queue size has the same distribution as the number of customers who arrive during a random time interval distributed as the stationary waiting time.

A Queueing Model for Road Traffic Flow

SUMMARY It is proposed that on roads which are uninterrupted by traffic signals, intersections, etc., vehicles should be considered as travelling in random queues. Criteria for determining the queues in actual traffic are found. A crude model is then used to study the formation of these queues in an attempt to derive the Borel–Tanner distribution of queue lengths. The random queues model is then used to study waiting times for pedestrians (or vehicles) wishing to cross one lane of traffic.