Erlang Formula Research Articles

Blocking probabilities in multitraffic loss systems: insensitivity, asymptotic behavior, and approximations

It is known that, under any sharing policy, the state describing the number of calls established for each class of traffic in steady state has a product-form distribution when the connection time distribution has a rational Laplace transform. The product-form property further holds for arbitrary holding time distribution under coordinate convex sharing policies. For the complete sharing policy case, an aggregate state describing the number of occupied circuits is shown to maintain the product-form property under asymptotic behavior, when the capacity and traffic intensities go to infinity on a comparable scale. Two theorems relative to the asymptotic behavior of the blocking probabilities which provide some insight into the nature of the blocking phenomenon are given. An approximation which reduces the numerical complexity of evaluating the blocking probabilities for the different classes of service to the computation of a single Erlang formula and the determination of the root of a monotonous polynomial function is proposed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Link access blocking in very large multi-media networks

This paper is concerned with performance models and approximations for very large, multi-media networks. Blocking is the traditional performance measure in circuit-switched networks, and will remain as such in multi-rate circuit-switched networks. It is also considered a likely performance measure candidate for resource management and dimensioning in broadband fast packet-switched networks, operated under virtual circuit mode. To gain some insight into network-wide blocking phenomena, we study in this paper the basic building block of any network, a single link. Multi-media traffic calls with Poisson arrivals, arbitrarily distributed service times, and different bandwidth requirements (in number of circuits), are offered to a system (link) of finite capacity. A call is rejected when the sharing policy doesn't allow it to be accommodated by the system. Under complete sharing policy, an aggregate state describing the number of occupied circuits is shown to exhibit a product form solution in steady state, when the capacity and traffic intensities become very large asymptotically. This structural property leads to the main result of this paper, an approximation which reduces the computation of the blocking probabilities for the different classes of traffic to the evaluation of a single Erlang formula and the determination of the root of a monotonous polynomial function. We provide comparison curves of the exact blocking probabilities and the approximations for some examples with 3 classes of traffic.

A generalization of the Erlang formula of traffic engineering

Calls arrive at a switch, where they are assigned to any one of the available idle outgoing links. A call is blocked if all the links are busy. A call assigned to an idle link may be immediately lost with a probability which depends on the link. For exponential holding times and an arbitrary arrival process we show that the conditional distribution of the time to reach the blocked state from any state, given the sequence of arrivals, is independent of the policy used to route the calls. Thus the law of overflow traffic is independent of the assignment policy. An explicit formula for the stationary probability that an arriving call sees the node blocked is given for Poisson arrivals. We also give a simple asymptotic formula in this case.

A stochastic interpretation of convective transport enhancement by external force fields

AbstractThe enhancing effect of external force fields on convection occurring at solid‐fluid interfaces is interpreted by means of a simple stochastic relationship derived from the Erlang formula in the theory of Markov processes.

Teletraffic Analysis for Single-Cell Mobile Radio Telephone Systems

Formulas are derived for the congestion in single-cell mobile radio systems in which there are both land-to-mobile and mobile-to-mobile calls and in which mobile-to-mobile calls go via the base station. Two approaches are used. The first yields modified forms of the familiar Erlang and Engset formulas. The second gives more complicated but more accurate formulas. The results of computer simulations to establish the accuracy of the formulas are described.

Overflow traffic from the viewpoint of renewal theory

SummaryIn this paper some known formulae concerning overflow traffic have been derived in a simpler way than is generally done in literature, i.e. by using simple results obtained from elementary renewal theory.After the model has been introduced, the distribution of intervals between overflows is calculated by means of a recurrence relation. A generalization of Erlang's formula for the call congestion follows almost immediately. The last section the special case of Poisson input is discussed.

The ergodicity of service systems with an infinite number of servomechanisms

Existence, uniqueness, and ergodicity are proved for a stationary distribution for a service system having countably many servomechanisms with input flow rate μk depending on the number k of servomechanisms occupied, and with arbitrary (identical) distribution of the service time with finite mean μ, under the condition\(\mu \mathop {\overline {\lim } }\limits_{k \to \infty } \frac{{\lambda _k }}{{k + 1}}< 1\). For this system we have, in particular, Erlang's formula $$p_k (t)\mathop \to \limits_{k + \infty } p_k = \frac{{\lambda _0 ...\lambda _{k - 1} \mu ^k }}{{k!}}p_0 ,k = 0,1,...,p_0^{ - 1} = \sum\nolimits_{k = 0}^\infty {\frac{{\lambda _0 ...\lambda _{k - 1} \mu ^k }}{{k!}}} ,\lambda _{ - 1} = 1.$$

An N-Server Stochastic Service System with Customer Preferences

The Erlang loss formula is currently used to estimate the capacity of a set of N linear curb spaces for vehicle parking; but this model applies only if drivers will accept the first vacant space they come to, a condition not found in practice. This paper develops a model of service that permits customers to decline an available server (space) when the walking distance would be excessive. The results enable the calculation of not only the overall loss, but the ability of the service system to satisfy customer preferences. It evaluates the conditional loss (given the space preferred), the access time, and the expected walking distance for each preference class. We give some examples of results based upon traffic characteristics forecast at a large, modern air terminal. It is found that the Erlang formula considerably underestimates the space requirements for linear curb parking and ignores the impact of its driver-indifference assumption upon passenger walking distance. The model has been programmed in APL for rapid calculations of all pertinent factors, allowing economical interactive use for planning—a curb with 30 spaces can be evaluated in approximately 1/2 second of CPU time.

SPECIAL CONSIDERATION ON THE DESIGN OF AN LNG HARBOR

One of the primary considerations in the design of an LNG harbor is safety, requiring berths to be separated by large distance and well protected from the outside wave agitation. Therefore, LNG harbors require expensive structures established as close as possible to the liquefaction plant (while crude tankers may be served by relatively much cheaper, single point mooring servers in deeper water). The cost of waiting time for the very expensive LNG ships has to be weighted against the cost of the additional berths and structures. (A 125, 000 m-^ LNG ships costs $2000/hour. ) The present paper describes the results of a study in which the optimum solution has been obtained by comparing these costs. The number of options is characterized by the number of berths. The cost to be added to the cost of construction of the berths includes an additional length of breakwater and additional dredging, plus the costs of financing during construction and the cost of maintenance. The waiting time for the LNG ship is generally determined, based on the classical Erlang formula for quelng theory. It is recalled that this formula is developed for an open loop. A closed loop theory has been specially developed for the present problem (since LNG ships will most probably operate between two well-defined harbors). The waiting times are 15 to 20% smaller than given by the closed loop theory. A comparison between single berth and double berth is examined. The effect of the rate of filling which is a function of the cryogenic pump capacity (or size of ship depending upon the dominating controlling factor) is analyzed. Finally, the sensitivity of the recommended solution as a function of the interest rates--examined in view of current economic uncertainties--is also investigated. The final recommendation for the design of the harbor, based on the prevailing factors, is the optimum economic solution.

Erlang's formula and some results on the departure process for a loss system

The present paper investigates the limiting distribution of the number of busy channels (queue size) and the remaining lengths of holding times at an epoch of departure for a loss system with general holding times and exponentially distributed interarrival times. Further, it is established that for this loss system in the limit an interdeparture interval length is independent of the queue size at the end of the interval and is distributed according to an exponential distribution with mean λ–1. It is also seen that in the limit interdeparture times are mutually independent.

Erlang's formula and some results on the departure process for a loss system

The present paper investigates the limiting distribution of the number of busy channels (queue size) and the remaining lengths of holding times at an epoch of departure for a loss system with general holding times and exponentially distributed interarrival times. Further, it is established that for this loss system in the limit an interdeparture interval length is independent of the queue size at the end of the interval and is distributed according to an exponential distribution with mean λ–1. It is also seen that in the limit interdeparture times are mutually independent.

Remark on a Service System with Mixed Input Streams

In a paper published in Opns. Res. 14, 145–156 (1966) J. P. Young discusses multiple-channel queuing systems with exponential service-time distribution, no waiting space, and two parallel input streams (Poisson input mixed with an L-phase Erlang input). This note discusses the consequences of an additional assumption introduced by Young to obtain steady-state probabilities of having n channels occupied. Since the assumption is justified only for the pure Poisson input in the case of mixed input, the Erlang formula (4), proposed by Young, gives only an approximation to the exact probabilities.

An extension of Erlang's formulas which distinguishes individual servers

For a particular kind of finite-server loss system in which the number and identity of servers depends on the type of the arriving call and on the state of the system, the limits of the state probabilities (as t → ∞) are found for an arbitrary service-time distribution.

An extension of Erlang's formulas which distinguishes individual servers

For a particular kind of finite-server loss system in which the number and identity of servers depends on the type of the arriving call and on the state of the system, the limits of the state probabilities (as t → ∞) are found for an arbitrary service-time distribution.

Das erlangsche model mit abhängigen bedienungszeiten

The service model of Erlang with n servers, Poisson input, and exponential distributed service times isgeneralized in such a way that one can consider service times which can be dependent and have arbitary distributions. A sufficient condition is given for the existence of a uniquely determined stationary distribution for the state of the system. As in the case of the usual Erlang model, this distribution can be described by the well known Erlang formulas. The condition requires that the service times of different servers form stationary series which are independent of each other.

On Erlang's Formula

On Erlang's Formula

On the generalization of Erlang's formula

On the generalization of Erlang's formula