A simple algebraic approximation to the Erlang loss system

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Abstract

We use Palm calculus to derive a simple, intuitive system of two linear-quadratic equations and two unknowns, whose algebraic solution yields Harel’s (1988) upper bound to the Erlang loss probability. We then derive a sequence of progressively stronger systems of equations, which eventually become exact. We provide two example applications.

Introduction

The Erlang loss system was originally studied by [3] in the context of telephony, and is one of the most thoroughly studied stochastic systems in the literature. Customers arrive according to a Poisson process with rate λ>0, and request allocation to any one of K1 service units indexed by kK{1,,K}. If no unit is available, the customer is lost. Otherwise, a unit is allocated and held by the customer for a service time having mean τ>0. Service times are independent and identically distributed according to a general distribution. In such systems, the Erlang loss probability, denoted by the function B(λτ,K), is of central interest because it measures the fraction of customers lost.

In recent years there has been considerable interest in optimal pricing and control of stochastic systems, inside which the Erlang loss system is embedded. Examples include pricing, capacity, and admissions control decisions in rental businesses, e.g. see [6], [9], and intermodal logistics [1]. Ideally, models for such settings should contain an accurate representation of Erlang loss performance, but in a parsimonious way that can be easily handled in a math programming model where other complex features are present.

One might consider directly using the Erlang loss formula, which measures the loss probability exactly, B(λτ,K)=(λτ)K/K!k=0K(λτ)k/k!. Alternatively, instead of the exact expression for B(λτ,K), one could use one of the many simpler bounds available. For example, Harel [4] gives a strong upper bound, called UP1, which is UP1=K(1ρ)2+2ρ(1ρ)4Kρ+K2(1ρ)2Kρ(1ρ)+2ρ+ρ4Kρ+K2(1ρ)2, where ρ=λτ/K is known as the traffic intensity. Other (weaker) bounds are given by Heyman [5] and Sobel [8]. With modern computers, both (1), (2) can be easily calculated, or approximated, for an Erlang loss system with fixed parameters. However, they can be too complex or even unworkable for direct use within mathematical programming models in which the parameters are either decision variables themselves, or nonlinear functions of other decision variables, such as in pricing applications. In some settings, such as multi-class admissions control, these formulas do not even fit the stochastic system, yet it is nevertheless essential to somehow capture Erlang loss-like behaviour.

In this paper, we use Palm calculus [2] to derive a nonlinear system of two equations and two unknowns, plus nonnegativity, whose solution in fact yields Harel’s upper bound (2). Relative to (1) or (2), this nonlinear system is extraordinarily simple and intuitive. The nonlinearity in the equations is quadratic (when K is fixed), which enables them to be handled much more simply than (1) or (2) in optimization models. We also extend these equations to obtain a sequence of higher-order approximations (though still only quadratic) that progressively approach the exact Erlang loss performance metrics. To our knowledge, ours is the only framework in the literature that leads to arbitrarily stronger upper bounds than UP1, and therefore the strongest upper bounds available. We report the first-order algebraic solution to these equations, which provides a completely new and more powerful formula than (2). Finally, we formulate optimization models for two applications.

Our analysis will use Palm calculus in Little’s Law, the PASTA (Poisson arrivals see time averages) property, and the following theorem. Suppose that {Y(t):tR} is a continuous-time stochastic process and Z is a point process of embedded epochs, which are jointly stationary on the same underlying probability space. Let P{}Z denote the Palm probability of an event occurring at an arbitrary epoch in Z. We will use the following result, e.g. see [2]:

Theorem 1 Superposition of Point Processes

Suppose that stationary point process Z , having intensity μ , is the superposition of stationary point processes Z1,Z2,,ZK , i.e. Z=Z1+Z2+ZK , where the process Zk has intensity μk . ThenP{}Z=k=1KμkμP{}Zk.

We denote the time stationary probability of an event by P{}. When the system is ergodic, Palm probabilities can be interpreted as empirical long-run event averages calculated at points of Z, whereas time stationary probabilities are empirical long-run time averages for the continuous-time process.

The key to our analysis is a novel view of the Erlang loss system. Until now, researchers have focused directly on the loss probability B(λτ,K) itself. In contrast, we consider the perspective of a single service unit. This makes analysis of the system amenable to Palm calculus. In particular, for each unit kK consider the stationary stochastic process {Yk(t)}tR, where Yk(t)=1 if unit k is available at time t, and 0 otherwise. Then we can define the time stationary probability that unit k is available as PkE[Yk(0)], where the expectation E is with respect to the underlying probability measure of the stochastic process. Similarly, we let Xk denote the intensity of the stationary point process that counts allocations of unit k. Little’s Law states that 1Pk=E[1Yk(0)]=τXk.

Next, we assume that all units experience the same statistical reality in the stationary system. That is, the labels of the units play no role in allocating them to requests when multiple units are available at once. An example of such an allocation rule is one that allocates the units in the order in which they become available (first-come-first-served), or one that chooses among the available units randomly. In general, when referring to a system having this property, we say that it has symmetric flows.

Under this assumption, neither Pk nor Xk depend on k. This allows us to drop the index k to work with the following basic quantities: for an arbitrary unit, P is the time stationary probability it is available, and X is the rate at which it is allocated. Hence, KX is the total effective throughput across all units, and KP is the total average number of units available for allocation. Furthermore, with the calculation of X in hand, the loss probability is merely 1KX/λ. Also, we will sometimes write P(K) and X(K) to emphasize that we are calculating P and X, respectively, for a system with K units.

Section snippets

Exact analysis

The following result provides an exact recursion for P(K). This result can alternatively be proved using the well-known recursion B(λτ,K)=λτB(λτ,K1)K+λτB(λτ,K1)K=1,2,, where B(λτ,0)1, and the fact that B(λτ,K)=1K(1P(K))λτ. (From Little’s Law, the effective throughput equals K(1P(K))τ.) However, what is of main interest here is our proof, which provides the framework and key steps for deriving our approximations.

Theorem 2

P(K)=11+λτ1+(K1)P(K1)K=1,2,,where P(0)=0 .

Proof

Let E denote the Poisson

Zeroth-order approximation: Harel’s upper bound

The proof of Theorem 2 shows that we can substitute PkDk=P(K1) into (4), which yields X=λP(K)(K1)XP(K1). The idea is to now approximate P(K1) with P(K). This is justified because Lemma 1 implies that {P(K)} is a Cauchy sequence, so that limK|P(K)P(K1)|=0. In other words, we replace a Palm probability (event average) with a time stationary probability (time average), and this is justified asymptotically as the number of units K increases. For notational convenience we drop the

Higher-order approximations

We next derive a series of K nonlinear systems that provide progressively stronger bounds, eventually yielding the exact analysis in Theorem 2. The idea behind the Nth-order system, where N=0,1,,K1, is to use (6) to approximate a system with KN units. Then, we effectively apply the recursion in (5) to compute performance measures for systems having KN+1 through K units, where this final calculation gives the desired bounds. As the solution to (6) exists and is unique, the solution after

Pricing and capacity sizing decisions

Consider a rental operation, in which a stock of K service units are maintained. When a customer arrives, he pays a price ζ to rent a unit for expected time τ. The aggregate arrival rate of customers is given by a demand function λ(ζ), which depends on the price. It costs the firm c whenever a customer rents a unit, and the firm also pays a holding cost per unit time of h for every unit K it maintains. The problem is to determine the stocking level K and price ζ so as to maximize the expected

Acknowledgements

The author thanks the University of Chicago Graduate School of Business for their financial support.

References (9)

  • S. Ziya et al.

    Optimal prices for finite capacity queueing systems

    Operations Research Letters

    (2006)
  • D. Adelman

    Price-directed control of a closed logistics queueing network

    Operations Research

    (2007)
  • F. Baccelli et al.

    Elements of Queueing Theory: Palm Martingale Calculus and Stochastic Recurrences

    (2003)
  • A.K. Erlang

    Solution of some problems in the theory of probabilities of significance in automatic telephone exchanges

    Electroteknikeren (Danish)

    (1917)
There are more references available in the full text version of this article.

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