Arie Harel

Convexity Properties of the Erlang Loss Formula

We prove that the throughput of the M/G/x/x system is jointly concave in the arrival and service rates. We also show that the fraction of customers lost in the M/G/x/x system is convex in the arrival rate, if the traffic intensity is below some Ii* and concave if the traffic intensity is greater than Ii*. For 18 or less servers, Ii* is less than one. For 19 or more servers, Ii* is between 1 and 1.5. Also, the fraction lost is convex in the service rate, but not jointly convex in the two rates. These results are useful in the optimal design of queueing systems.

Strong Convexity Results for Queueing Systems

We prove a strong and seemingly odd result about the M/M/c queue: the reciprocal of the average sojourn time is a concave function of the traffic intensity. We use this result to show that the average itself is jointly convex in arrival and service rates. The standard deviation has the same properties. Also, we determine conditions under which these properties are exhibited by a standard approximation for the M/G/c queue. These results are useful in design studies for telecommunications and production systems.

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.

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.

CONVEXITY RESULTS FOR SINGLE-SERVER QUEUES AND FOR MULTISERVER QUEUES WITH CONSTANT SERVICE TIMES

We show that the waiting time in queue and the sojourn time of every customer in the GIGI1 and GIDIc queue are jointly convex in mean interarrival time and mean service time, and also jointly convex in mean interarrival time and service rate. Counterexamples show that this need not be the case, for the GIIGI/c queue or for the DIGIIc queue, for c 2 2. Also, we show that the average number of customers in the MID/c queue is jointly convex in arrival and service rates. These results are surprising in light of the negative result for the GIIGI/2 queue

Simple bounds for closed queueing networks

Consider a closed Jackson type network in which each queue has a single exponential server. Assume that N customers are moving among k queues. We establish simple closed form bounds on the network throughput (both lower and upper), which are sharper than those that are currently available. Numerical evaluation indicates that the improvements are significant.

Random walk and the area below its path

A simple random walk with reflected origin is considered. The walk starts at the origin and it must return to the origin at time 2n. We show that the expected area below the path of this walk is n22n−1/Cn2n. If, however, the walk is required to return to the origin for the first time at time 2n, then the expected area below the path of this Bernoulli excursion is (2n − 1)22n−1/Cn2n. We also show that if V1 < ⋯ < Vn is the order statistics based on a sample of size n from a uniform distribution over (0, 1), and that if U1 < ⋯ < Un is another independent set of order statistics from the same distribution, then E [∑i=1n |Vi − Ui|] = n22n−1/[(2n + 1)Cn2n] We use this result to find an average-case performance of the Earliest Due Date (EDD) heuristic for one machine scheduling problem with earliness and tardiness penalties. We also apply some of the results to Larsons Queue Inference Engine (1990).

Project Valuation With Time-Varying Cash Flows: A Bayesian Approach

This article investigates the valuation of a project when the distributions of cash flows vary over time. The decision maker is assumed to be a Bayesian decision maker under uncertainty. Using the dynamic programming principle of backward induction and assuming that the capital asset pricing model is valid in each time period, we derive the projects valuation formulas and systematic risks, and investigate their characteristics. Our valuation formulas embed a Bayesian learning effect and differ from the traditional textbook capital budgeting formulas.

Pricing Securities With Exchange-Imposed Price Limits Via Risk Neutral Valuation

Asian and European financial markets impose daily price fluctuation limits on individual securities. In the US several futures exchanges are regulated by price fluctuation limits as well. The price limits in most exchanges are set daily, and they are usually based on a percentage change from the previous days closing price. We show that the future cash flows of a security subject to price limit regulation resemble that of a distinctive contingent claim. Assuming that the security price follows a lognormal distribution, we use the risk-neutral valuation relation (RNVR) developed by [4] to derive the security valuation, in the presence of price fluctuation limits. The characteristics of the pricing formula are examined analytically and numerically.

SECURITY MARKETS WITH PRICE LIMITS: A BAYESIAN APPROACH

Several financial markets impose daily price limits on individual securities. Once a price limit is triggered, investors observe either the limit floor or ceiling, but cannot know with certainty what the true equilibrium price would have been in the absence of such limits. The price limits in most exchanges are typically based on a percentage change from the previous days closing price, and can be expressed as return limits. We develop a Bayesian forecasting model in the presence of return limits, assuming that security returns are governed by identically and independently shifted-exponential random variables with an unknown parameter. The unique features of our Bayesian model are the derivations of the posterior and predictive densities. Several numerical predictions are generated and depicted graphically. Our main theoretical result with policy implications is that when return-limit regulations are tightened, the price-discovery process is impeded and investors welfare is reduced.