Train Arrivals Research Articles

Statistical multiplexing of correlated variable-length packet trains: an analytic performance study

We consider a statistical multiplexer which is modeled as a discrete-time single-server queueing system. Messages consisting of a variable number of fixed-length packets arrive to the muliplexer at the rate of one packet per slot (‘train arrivals’), which results in what we call a primary correlation in the packet arrival process. The distribution of the message lengths (in terms of packets) is general. Additionally, the arrival process exhibits a secondary correlation, which results from the fact that the distribution of the number of leading packet arrivals in a slot depends on some environment variable. We assume this environment to have two possible states, each with geometric sojourn times. By using generating functions and an infinite-dimensional state description, we derive closed-form expressions for the mean, the variance and the tail distribution of the buffer contents in the steady state. Some numerical examples illustrate the effect of both primary and secondary correlation on the multiplexer performance.

Ex ante heuristic measures of schedule reliability

Measures of reliability and punctuality of scheduled public transport services are important in planning, management, operating and marketing of these services. Various methods can be used to measure reliability. Analytic methods are usually practical for only very simple structured systems. Simulation methods are very time consuming and require data which may not be available. As a result, the most widely used measures are ad hoc or heuristic. However, the assumptions and properties of these measures, and the relationships between them, are seldom discussed, hence we discuss them here. We consider existing measures, extensions of these, and new measures. For specificity, we use the example of train arrivals and departures at a train station: stations with several hundred trains per day and multiple platforms are common in many countries, for example throughout Europe. Some measures of reliability are based on the observed delays, hence can be used only after the event. However, we here focus on measures which can be used in advance, for example for estimating the reliability of proposed schedules or changes in schedules at the design stage. In this we distinguish between measures which require some information about probabilities of delay and those which do not. We also distinguish between exogenous delays, which are beyond the influence of the scheduler (delays due to problems in engineering or operations), and knock-on delays which are affected by schedule design: both types are of interest for schedule reliability, but the latter are of more interest for measuring schedule robustness.

Correlation effects in ATM queues due to data format conversions

We study the performance of a statistical multiplexer to which messages composed of a variable number of fixed-length packets arrive at the rate of one packet per slot (“train arrivals”), resulting in a correlated packet arrival stream. In this paper, unlike in previous related studies, the distribution of the message lengths is general. We introduce a technique for the exact analysis of the system, which is essentially a generating-functions approach that uses an infinite-dimensional state description. Explicit expressions are obtained for the probability generating functions of the queue length and the packet delay. Also, closed-form results are found for the mean, variance and tail distribution of the queue length and the packet delay. The mean message waiting time and the mean message delay are derived as well. A comparison is made between the buffer behavior with train arrivals and that with batch arrivals, where all the packets of a message enter the buffer during the same slot, and also it is shown that the correlation has a strong effect on the performance of the system. By means of some numerical examples, we also show that the exact nature of the message-length distribution has a significant impact on the multiplexer performance.

Discrete-time buffers with variable-length train arrivals

An infinite capacity single server queueing system is considered, to which messages consisting of a variable number of fixed-length packets arrive at the rate of one packet per slot (‘train arrivals’). Assuming geometrically distributed message lengths, an exact closed-form expression for the probability generating function (pgf) of the buffer contents is obtained.

Priority queueing system with fixed-length packet-train arrivals

A discrete-time priority queueing system is studied, in which two different classes of fixed-length packet-trains arrive according to independent batch geometric streams. The packets in each packet-train arrive at the rate of one packet per slot (train arrivals), resulting in a correlated arrival stream. The service time of one packet is deterministic of one slot. The motivation for the work comes from ATM networks with diverse traffic sources and correlated packet arrival stream. Using the probability generating function method, the joint distribution of queue lengths and the waiting time distribution are obtained for each class. Numerical results are presented. Comparision is made with the case of ‘batch arrivals’, where all packets of a packet-train arrive simultaneously at the buffer.

Output‐queue contents in a ‘minislot’ synchronized ATM switch

AbstractAn N × N switching element with output queueing, as used in a large ATM switching network, is considered. All the inlets of the switching element are synchronized on ‘minislots’, where a minislot is the fixed‐length time unit for the transmission of one ‘minicell’. When entering the switching network, an ATM cell is converted into a ‘minicell‐train’, consisting of a fixed number of minicells. Using an active/silent model, it is assumed that on each inlet of the switching element, the number of minicelltrains in an active period and the length of a silent period are both geometrically distributed, and the arriving minicell‐trains are uniformly distributed among all the outlets. The performance of the switching element can be obtained by analysing one single output queue, which is modeled as a discrete‐time single‐server queuing system with train arrivals. In this paper, an upper bound and an approximate expression for the mean queue length are derived. More importantly, an analytical method is developed to obtain a tight upper bound and a good approximation for the tail distribution of the queue length. This analytical method is very useful in buffer dimensioning of ATM switches.

Buffer contents and delay for statistical multiplexers with fixed-length packet-train arrivals

A slotted statistical multiplexer is considered, to which messages containing a fixed number of fixed-length packets each, arrive at the rate of one packet per slot. The multiplexer buffer can therefore be studied using a discrete-time queueing model with correlated arrivals, referred to as “train arrivals” in the paper. This train arrival model is completely analyzed by means of a generating-functions approach. An explicit expression for the probability generating function (pgf) of the buffer contents is obtained. Furthermore, a simple relationship is derived between the pgf of the buffer contents and the pgf of the packet delay in a discrete-time G/D/1 queueing system, which is, of course, also valid for the train arrivals case. Finding this simple and general relationship is one of the contributions of this paper. The pgf of the message delay is obtained as well. Some numerical results are presented. A comparison is also made with the case of “batch arrivals” in which all packets of a message arrive simultaneously to the buffer.

Optimal slack time for timed transfers at a transit terminal

AbstractAt transit terminals where two routes interchange passengers, total system costs may be reduced by allowing some “slack” time in the vehicle schedules to decrease the probability of missed connections. Transfer cost functions are formulated and used to determine optimal slack time for simple systems with transfers between one bus route and one rail line. Some analytic results are derived for empirical discrete and Gumbel distributions of bus arrival times. Relations between the optimal slack times and headways, transfer volumes, passenger time values, bus operating costs, and standard deviations of bus and train arrivals are also developed numerically using normally distributed arrivals. However, the proposed numerical approach can optimize slack times for any observed arrival distributions. The results provide some guidelines on desirable slack times and show that schedule coordination between the two routes is not worth attempting when standard deviations of arrivals exceed certain levels. Possible extensions of this work are suggested in the last section.

Exiting: Another Look

Confirmation of a prior study showing students preferred to exit via an open door was based on choice of an open door by most of about 102 persons per 201 train arrivals observed on 25 unobtrusive weekday observations.

EXITING: ANOTHER LOOK

Confirmation of a prior study showing students preferred to exit via an open door was based on choice of an open door by most of about 102 persons per 201 train arrivals observed on 25 unobtrusive weekday observations.

Packet Trains--Measurements and a New Model for Computer Network Traffic

Traffic measurements on a ring local area computer network at the Massachusetts Institute of Technology are presented. The analysis of the arrival pattern shows that the arrival processes are neither Poisson nor compound Poisson. An alternative model called "packet train" is proposed. In the train model, the traffic on the network consists of a number of packet streams between various pairs of nodes on the network. Each node-pair stream (or node-pair process, as we call them) consists of a number of trains. Each train consists of a number of packets (or cars) going in either direction (from node A to B or from node B to A). The intercar gap is large (compared to packet transmission time) and random. The intertrain time is even larger. The Poisson and the compound Poisson arrivals are shown to be special cases of the train arrival model. Another important observation is that the packet arrivals exhibit a "source locality." If a packet is seen on the network going from A to B, the probability of the next packet going from A to B or from B to A is very high. Implications of the train arrivals and of source locality on the design of bridges, gateways, and reservation protocols are discussed. A numbet of open problems requiring development of analysis techniques for systems with train arrival processes are also described.