Yoav Kerner

The Conditional Distribution of the Residual Service Time in the M n /G/1 Queue

We study a single server queueing system with general service time distribution and memoryless inter-arrival times. The arrival rate is not constant but varies with the number of customers in the system. A recursive formula for the conditional distribution of the remaining service time given the queue length is derived for an arbitrary epoch. It is also shown that this conditional distribution holds for arrival epochs. The recursion for the corresponding Laplace–Stieltjes transforms and expected values is given in a simple formula.

On balking from an empty queue

Abstract The intuition while observing the economy of queueing systems, is that one’s motivation to join the system, decreases with its level of congestion. Here we present a queueing model where sometimes the opposite is the case. The point of departure is the standard first-come first-served single server queue with Poisson arrivals. Customers commence service immediately if upon their arrival the server is idle. Otherwise, they are informed if the queue is empty or not. Then, they have to decide whether to join or not. We assume that the customers are homogeneous and when they consider whether to join or not, they assess their queueing costs against their reward due to service completion. As the whereabouts of customers interact, we look for the (possibly mixed) join/do not join Nash equilibrium strategy, a strategy that if adopted by all, then under the resulting steady-state conditions, no one has any incentive not to follow it oneself. We show that when the queue is empty then depending on the service distribution, both ‘avoid the crowd’ (ATC) and ‘follow the crowd’ (FTC) scenarios (as well as none-of-the-above) are possible. When the queue is not empty, the situation is always that of ATC. Also, we show that under Nash equilibrium it is possible (depending on the service distribution) that the joining probability when the queue is empty is smaller than it is when the queue is not empty.

Pricing Time-Sensitive Services Based on Realized Performance

Services such as FedEx charge up-front fees but reimburse customers for delays. However, lead-time pricing studies ignore such delay refunds. This paper contributes to filling this gap. It studies revenue-maximizing tariffs that depend on realized lead times for a provider serving multiple time-sensitive customer types. We relax two key assumptions of the standard model in the lead-time pricing literature. First, customers may be risk averse RA with respect to payoff uncertainty, where payoff equals valuation, minus delay cost, minus payment. Second, tariffs may be arbitrary functions of realized lead times. The standard model assumes risk-neutral RN customers and restricts attention to flat rates. We report three main findings: 1 With RN customers, flat-rate pricing maximizes revenues but leaves customers exposed to payoff variability. 2 With RA customers, flat-rate pricing is suboptimal. If types are distinguishable, the optimal lead-time-dependent tariffs fully insure delay cost risk and yield the same revenue as under optimal flat rates for RN customers. With indistinguishable RA types, the differentiated first-best tariffs may be incentive-compatible even for uniform service, yielding higher revenues than with RN customers. 3 Under price and capacity optimization, lead-time-dependent pricing yields higher profits with less capacity compared to flat-rate pricing.

Equilibrium strategies in queues based on time or index of arrival

In most decision models dealing with unobservable stochastic congested environments, one looks for a (Nash) equilibrium behavior among customers. This is a strategy that, if adopted by all, then under the resulting steady-state conditions; the best response for an individual is to adopt this strategy too. The purpose of this article is to look for a simple decision problem but where the assumption of steady-state conditions is removed. Specifically, we consider an M/M/N/N loss model in which one pays for trying to get service but is rewarded only if one finds an available server. The initial conditions at time 0 are common knowledge and each customer possesses his arrival time as his private information. The equilibrium profile tells each arrival whether to try (randomization allowed) given his time of arrival. We show that all join up to some point of time. At this point, there is a quantum drop in the joining probability from one to some fraction. From then on, their joining probability continuously converges to the equilibrium joining probability under the model that assumes steady state.

The age of the arrival process in the G/M/1 and M/G/1 queues

This paper shows that in the G/M/1 queueing model, conditioning on a busy server, the age of the inter-arrival time and the number of customers in the queue are independent. The same is the case when the age is replaced by the residual inter-arrival time or by its total value. Explicit expressions for the conditional density functions, as well as some stochastic orders, in all three cases are given. Moreover, we show that this independence property, which we prove by elementary arguments, also leads to an alternative proof for the fact that given a busy server, the number of customers in the queue follows a geometric distribution. We conclude with a derivation for the Laplace Stieltjes Transform (LST) of the age of the inter-arrival time in the M/G/1 queue.

The intercept term of the asymptotic variance curve for some queueing output processes

We consider the output processes of some elementary queueing models such as the M/M/1/K queue and the M/G/1 queue. An important performance measure for these counting processes is their variance curve v(t), which gives the variance of the number of customers in the time interval [0, t]. Recent work has revealed some non-trivial properties dealing with the asymptotic rate at which the variance curve grows. In this paper we add to these results by finding explicit expressions for the intercept term of the linear asymptote. For M/M/1/K queues our results are based on the deviation matrix of the generator. It turns out that by viewing output processes as Markovian Point Processes and considering the deviation matrix, one can obtain explicit expressions for the intercept term, together with some further insight regarding the BRAVO (Balancing Reduces Asymptotic Variance of Outputs) effect. For M/G/1 queues our results are based on a classic transform of D. J. Daley. In this case we represent the intercept term of the variance curve in terms of the first three moments of the service time distribution. In addition we shed light on a conjecture of Daley, dealing with characterization of stationary M/M/1 queues within the class of stationary M/G/1 queues, based on the variance curve.

A queueing approach to a multi class M/G/1 make-to-stock with backlog

This paper studies an M/G/1 production system serving several customer classes. We show that the Multilevel Rationing (MR) policy, that has been shown to be optimal in the M/M/1 case is not optimal in general. We propose another policy, which we call the extended MR (EMR). The EMR policy exploits the information on the number of all waiting customers at arrival epochs to assess the residual service time. We establish conditions under which the EMR policy is optimal and extend the conditions under which the MR policy is optimal.

Sonographic evaluation of intra‐abdominal adhesions during the third trimester of pregnancy: a novel technique in women undergoing repeated cesarean section

Intra‐abdominal adhesions are associated with an increased risk of complications during repeat Cesarean section (CS), such as bladder and bowel injury, hemorrhage, infection and hysterectomy. We present a simple sonographic marker, the ‘sliding sign’ of the uterus, for the prediction of intra‐abdominal adhesions in the third trimester of pregnancy in women undergoing repeat CS.

On non-equilibria threshold strategies in ticket queues

In many real-life queueing systems, a customer may balk upon arrival at a queueing system, but other customers become aware of it only at the time the balking customer was to start service. Naturally, the balking is an outcome of the queue length, and the decision is based on a threshold. Yet the inspected queue length contains customers who balked. In this work, we consider a Markovian queue with infinite capacity and with customers that are homogeneous with respect to their cost reward functions. We show that that no threshold strategy can be a Nash equilibrium strategy. Furthermore, we show that for any threshold strategy adopted by all, the individual’s best response is a double threshold strategy. That is, join if and only if one of the following is true: (i) the inspected queue length is smaller than one threshold, or (ii) the inspected queue length is larger than a second threshold. Our model is under the assumption that the response time of the server when he finds out that a customer balked is negligible. We also discuss the validity of the result when the response time is not negligible.

Optimization of Waiting Time in H-R Coordination

An analytical model of Human-Robot (H-R) coordination is presented for a Human-Robot system executing a collaborative task in which a high level of synchronization among the agents is desired. The influencing parameters and decision variables that affect the waiting time of the collaborating agents were analyzed. The performance of the model was evaluated based on the costs of the waiting times of each of the agents at the pre-defined spatial point of handover. The model was tested for two cases of dynamic H-R coordination scenarios. Results indicate that this analytical model can be used as a tool for designing an H-R system that optimizes the agent waiting time thereby increasing the joint-efficiency of the system and making coordination fluent and natural.