Erlang Delay Research Articles

Parisian Ruin with Erlang Delay and a Lower Bankruptcy Barrier

Parisian ruin occurs once the surplus stays continuously below zero for a given period. We consider the spectrally negative Levy risk process where ruin is declared either at the first time that the reserve stays continuously below zero for an exponentially or mixed Erlang distributed random variable, or once it reaches a given negative threshold. We consider the Laplace transform of the time to ruin and the Laplace transform of the time that the process is negative.

Resource Pooling and Cost Allocation Among Independent Service Providers

We study a situation where several independent service providers collaborate by complete pooling of their resources and customer streams into a joint service system. These service providers may represent such diverse organizations as hospitals that pool beds or maintenance firms that pool repairmen. We model the service systems as Erlang delay systems (M/M/s queues) that face a fixed cost rate per server and homogeneous delay costs for waiting customers. We examine rules to fairly allocate the collective costs of the pooled system amongst the participants by applying concepts from cooperative game theory. We consider both the case where players’ numbers of servers are exogenously given and the scenario where any coalition picks an optimal number of servers. By exploiting new analytical properties of the continuous extension of the classic Erlang delay function, we provide sufficient conditions for the games under consideration to possess a core allocation (i.e., an allocation that gives no group of players an incentive to split off and form a separate pool) and to admit a population monotonic allocation scheme (whereby adding extra players does not make anyone worse off). This is not guaranteed in general, as illustrated via examples.

Internet and the Erlang formula

We demonstrate that the Internet has a formula linking demand, capacity and performance that in many ways is the analogue of the Erlang loss formula of telephony. Surprisingly, this formula is none other than the Erlang delay formula. It provides an upper bound on the probability a flow of given peak rate suffers degradation when bandwidth sharing is max-min fair. Apart from the flow rate, the only relevant parameters are link capacity and overall demand. We explain why this result is valid under a very general and realistic traffic model and discuss its significance for network engineering.

In This Issue

In This Issue

Convexity Results for the Erlang Delay and Loss Formulae When the Server Utilization Is Held Constant

This paper proves a long-standing conjecture regarding the optimal design of the M/M/s queue. The classical Erlang delay formula is shown to be a convex function of the number of servers when the server utilization is held constant. This means that when the server utilization is held constant, the marginal decrease in the probability that all servers are busy in the M/M/s queue brought about by the addition of two extra servers is always less than twice the decrease brought about by the addition of one extra server. As a consequence, a method of marginal analysis yields the optimal number of servers that minimize the waiting and service costs when the server utilization is held constant. In addition, it is shown that the expected number of customers in the queue and in the system, as well as the expected waiting time and sojourn in the M/M/s queue, are convex in the number of servers when the server utilization is held constant. These results are useful in design studies involving capacity planning in service operations. The classical Erlang loss formula is also shown to be a convex function of the number of servers when the server utilization is held constant.

Contributors

Contributors

Service Level Approximations for a Call Center with Two Customer Types

We consider a call center whose customers can be classified into two types depending on their tendency towards waiting in the queue. A customer of type A waits in the queue, if necessary, until he gets served. On the other hand, a type B customer is assumed to be impatient and he does not join the queue and leaves the call center immediately if there is no server available. There are well known formulas for calculating the service levels when a call center serves only one customer type; Erlang delay formula and Erlang loss formula for type A and type B customers, respectively. Hence, there is no such simple analytic result for computing service levels when a call center handles both customer types together. In this paper we suggest and compare several approximation methods to calculate type A and B customer service levels for a call center serving both customer types. Numerical experiments show that one of the methods suggested in this paper approximates closely the actual service level especially when the call center provides high level of service

A lower bound for the Erlang C formula in the Halfin–Whitt regime

One of the classical models of queueing theory is the M/M/s queue or Erlang delay model. This model has s homogeneous servers working in parallel. Customers arrive according to a Poisson process with arrival rate λ, and the service times are independent and exponentially distributed with mean 1/μ. Let the offered load be a = λ/μ and assume a < s to have a proper steady-state distribution. The most important performance characteristic for this system is the probability that a customer is delayed when arriving at the system in steady state. This probability is known as the Erlang C formula, given by (with ρ = a/s)

Sharp and simple bounds for the Erlang delay and loss formulae

We prove some simple and sharp lower and upper bounds for the Erlang delay and loss formulae and for the number of servers that invert the Erlang delay and loss formulae. We also suggest simple and sharp approximations for the number of servers that invert the Erlang delay and loss formulae. We illustrate the importance of these bounds by using them to establish convexity proofs. We show that the probability that the M/M/s queue is empty is a decreasing and convex function of the traffic intensity. We also give a very short proof to show that the Erlang delay formula is convex in the traffic intensity when the number of servers is held constant. The complete proof of this classical result has never been published. We also give a very short proof to show that the Erlang delay formula is a convex function of the (positive integer) number of servers. One of our results is then used to get a sharp bound to the Flow Assignment Problem.

Heavy-traffic extreme value limits for Erlang delay models

We consider the maximum queue length and the maximum number of idle servers in the classical Erlang delay model and the generalization allowing customer abandonment--the M/M/n+M queue. We use strong approximations to show, under regularity conditions, that properly scaled versions of the maximum queue length and maximum number of idle servers over subintervals [0,t] in the delay models converge jointly to independent random variables with the Gumbel extreme value distribution in the quality-and-efficiency-driven (QED) and ED many-server heavy-traffic limiting regimes as n and t increase to infinity together appropriately; we require that t n ?? and t n =o(n 1/2?? ) as n?? for some ?>0.

Determining the Proper Number and Price of Software Licenses

Software houses sell their products by transferring usage licenses of various software components to the customers. Depending on the kind of software, there are several different license types that allow controlled access of services. The two most popular types are the fixed license, which gives access rights for an identified workstation, and the floating license, which restricts the number of simultaneous users to a certain bound. The latter of these types is advantageous when the users do not demand full-time services and occasional lack of access is bearable. The problem of deciding the number of floating licenses is studied in the present paper. Based on the expected usage profile of the software, we calculate the minimal number of licenses that guarantees that the customers get service better than a given lower bound. The problem is studied by using certain queuing models, known as the Erlang toss system, the Erlang delay system, and the Engset model. None of these analytic models consider, however, the transient period that we analyze by means of simulation and by the so-called modified offered load approximation. We also give simple formulas presenting how the number of software licenses needed to keep the probability of nonaccess below a given blocking level grows as a function of the offered load, which is the proportion of the time used in the case that all requests were successful. Results of the study may be used for setting license prices and for determining the proper number of licenses.

Bandwidth Estimation for Best-Effort Internet Traffic

A fundamental problem of Internet traffic engineering is bandwidth estimation: determining the bandwidth (bits per second) required to carry traffic with a specific bit rate (bits per second) offered to an Internet link and satisfy quality-of-service requirements. The traffic is packets of varying sizes that arrive for transmission on the link. Packets can queue up and are dropped if the queue size (bits) is bigger than the size of the buffer (bits) for the queue. For the predominant traffic on the Internet, best-effort traffic, quality metrics are the packet loss (fraction of lost packets), a queueing delay (seconds) and the delay probability (probability of a packet exceeding the delay). This article presents an introduction to bandwidth estimation and a solution to the problem of best-effort traffic for the case where the quality criteria specify negligible packet loss. The solution is a simple statistical model: (1) a formula for the bandwidth as a function of the delay, the delay probability, the traffic bit rate and the mean number of active host-pair connections of the traffic and (2) a random error term. The model is built and validated using queueing theory and extensive empirical study; it is valid for traffic with 64 host-pair connections or more, which is about 1 megabit/s of traffic. The model provides for Internet best-effort traffic what the Erlang delay formula provides for queueing systems with Poisson arrivals and i.i.d. exponential service times.

INSIGHTS ON SERVICE SYSTEM DESIGN FROM A NORMAL APPROXIMATION TO ERLANG'S DELAY FORMULA

We show how a simple normal approximation to Erlang's delay formula can be used to analyze capacity and staffing problems in service systems that can be modeled as M/M/s queues. The numbers of servers, s, needed in an M/M/s queueing system to assure a probability of delay of, at most, p can be well approximated by s ≅ p + z ***I‐ p+, where z 1‐ p, is the (1 ‐ p)th percentile of the standard normal distribution and ρ, the presented load on the system, is the ratio of Λ, the customer arrival rate, to μ, the service rate. We examine the accuracy of this approximation over a set of parameters typical of service operations ranging from police patrol, through telemarketing to automatic teller machines, and we demonstrate that it tends to slightly underestimate the number of servers actually needed to hit the delay probability target—adding one server to the number suggested by the above formula typically gives the exact result. More importantly, the structure of the approximation promotes operational insight by explicitly linking the number of servers with server utilization and the customer service level. Using a scenario based on an actual teleservicing operation, we show how operations managers and designers can quickly obtain insights about the trade‐offs between system size, system utilization and customer service. We argue that this little used approach deserves a prominent role in the operations analyst's and operations manager's toolbags.

Approximations for the delay probability in the M/G/s queue

This paper develops approximations for the delay probability in an M / G / s queue. For M / G / s queues, it has been well known that the delay probability in the M / M / s queue, i.e., the Erlang delay formula, is usually a good approximation for other service-time distributions. By using an excellent approximation for the mean waiting time in the M / G / s queue, we provide more accurate approximations of the delay probability for small values of s. To test the quality of our approximations, we compare them with the exact value and the Erlang delay formula for some particular cases.

A note on the convexity of the expected queue length of the M/M/s queue with respect to the arrival rate: a third proof

The convexity of the expected number in an M/M/s queue with respect to the arrival rate (or traffic intensity) is well known. Grassmann [1] proves this result directly by making use of a bound on the probability that all servers are busy. Independently, Lee and Cohen [2] derive this result by showing that the Erlang delay formula is a convex function. In this note, we provide a third method of proof, which exploits the relationship between the Erlang delay formula and the Poisson probability distribution. Several interesting intermediate results are also obtained.

Convexity of Functions which are Generalizations of the Erlang Loss Function and the Erlang Delay Function

Convexity of Functions which are Generalizations of the Erlang Loss Function and the Erlang Delay Function

Convexity of Functions Which are Generalizations of the Erlang Loss Function and the Erlang Delay Function

Convexity of Functions Which are Generalizations of the Erlang Loss Function and the Erlang Delay Function

The convexity of loss rate in an Erland loss system and sojourn in an Erlang delay system with respect to arrival and service rates

The Erlang loss system is a system in which customers arriving in a Poisson stream of arrival rate lambda are served by a group of n servers, with a mean service time of 1/ mu . Arrivals that find all servers busy are blocked and cleared from the system. The Erlang delay system is a system in which customers who find all servers busy wait in a queue until served by the first available server. Proofs of joint-convexity results in the Erlang delay system and Erlang loss system are presented, using induction on the number of servers and making use of the relation between the Erlang B and the Erlang C functions. The approach leads to a straightforward algebraic analysis of the properties of the performance measures of interest and avoids cumbersome manipulations.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Erratum to: Sharp Bounds and Simple Approximations for the Erlang Delay and Loss Formulas

Erratum to Sharp Bounds and Simple Approximations for the Erlang Delay and Loss Formulas (Vol. 34, No. 8 (August 1988), pp. 959–972.)

Sharp Bounds and Simple Approximations for the Erlang Delay and Loss Formulas

Some sharp bounds and simple approximations are obtained for the Erlang delay formula and for the Erlang loss formula. One of the results is then used to get a simple analytical solution for the Server Allocation Problem.