Large Arrival Rate Research Articles

Optimal pricing policies for tandem queues: Asymptotic optimality

We study the optimal pricing problem for a tandem queueing system with an arbitrary number of stations, finite buffers, and blocking. The problem is formulated using a Markov decision process model with the objective to maximize the long-run expected time-average revenue or gain of the service provider. Our interest lies in comparing the performances of static and dynamic pricing policies in maximizing the gain. We show that the optimal static pricing policies perform as well as the optimal dynamic pricing policies when the buffer size at station 1 becomes large and the arrival rate is either small or large. More importantly, we propose two specific static pricing policies for systems with small and large arrival rates, respectively, and show that each proposed policy produces a gain converging to the optimal gain with an approximately exponential rate as the buffer size before station 1 becomes large. We learn from numerical results that the proposed static policies perform as well as optimal dynamic policies even for a moderate-sized buffer at station 1. We also learn that there exist cases where optimal static pricing policies are, however, neither optimal nor near-optimal.

A central limit theorem for a Markov-modulated infinite-server queue with batch Poisson arrivals and binomial catastrophes

This paper considers the stationary queue length distribution of a Markov-modulated MX/M/∞queue with binomial catastrophes. When a binomial catastrophe occurs, each customer is either removed with a probability or is retained with the complementary probability. We focus on our model under a heavy traffic regime because its exact analysis is difficult if not impossible. We establish a central limit theorem for the stationary queue length of our model in a heavy traffic regime. The central limit theorem can be used to approximate the queue length distribution of our model with large arrival rates.

A central limit theorem for a Markov-modulated infinite-server queue with batch Poisson arrivals and binomial catastrophes

This paper considers the stationary queue length distribution of a Markov-modulated MX/M/∞ queue with binomial catastrophes. Binomial catastrophes occur according to a Poisson process, and each customer is removed with a probability and retained with the complementary probability upon the arrival of a binomial catastrophe. We focus on our model under a heavy traffic regime because its exact analysis is difficult if not impossible. We establish a central limit theorem for the stationary queue length of our model in the heavy traffic regime. The central limit theorem can be used to approximate the queue length distribution of our model with large arrival rates.

Coasian Bargaining with an Arriving Outside Option

We consider Coasian bargaining problems where the buyer has an outside option arriving at a stochastic time. We study both observable outside option models and unobservable outside option models. In both models, we show that a Coasian equilibrium exists if (1) the arrival of the outside option is public, or (2) the arrival of the outside option is private but the arrival probability is small enough. (1) the seller makes multiple rounds of offers, and (2) the Coase conjecture holds for an arbitrarily large arrival rate of the outside option. The result also applies to the time-varying outside option model. This exercise helps us to understand the sharp difference between Board and Pycia (2013), where the buyer's outside option is always available, and the standard Coasian bargaining literature, where the buyer has no outside option.

Study of the Effects of Multibit Error Correction Codes on the Reliability of Memories in the Presence of MBUs

Failures produced by soft errors in memories because of radiation or similar causes are an important problem due to the large impact that this issue has on the systems where they operate. There are several techniques that offer good results against single event upsets (SEUs), like single error correction-double error detection codes. However, there may be situations where these solutions are not enough, for example, when there is a large SEU arrival rate or multiple bit upsets (MBUs) are present. This last situation is becoming more frequent as technology scales, since in this case, the probability that a single event induces several errors in adjacent cells is higher. A possible solution to address this problem is the use of multibit protection codes, which are often utilized in other fields like communications. However, there are no developed models for memories that can put in perspective how these multibit protection codes behave in the presence of MBUs, and how the reliability of the system is affected by its use. In this paper, a reliability analysis of this case will be presented, studying how the proposed model behaves with respect to the traditional SEU ones and offering some applicability conditions. Both scenarios of memories with and without scrubbing will be analyzed, and several simulation results will be offered in order to prove the accuracy of the proposals.

Safe peer-to-peer self-downloading

A goal of peer-to-peer applications is to share files between users themselves rather than downloading files from file servers. Self-downloading protocols have the property that, eventually, every user downloads only from other users. Self-downloading is problematic if users disconnect from the system upon completing file downloading, because they only share with other users while connected. Yet, if users continue to arrive at a sufficient rate, self-downloading protocols are possible. One vulnerability of file sharing between users is the possibility that files or segments could be counterfeit or corrupt. Protocols that are d -safe tolerate some number of instances of faulty segments in a file being downloaded, because each segment is downloaded d times before being shared. This article shows that d -safe self-downloading is possible for a sufficiently large arrival rate of users to the system. Upper and lower connectivity and sharing bounds are given for d = 2, and simulation results show effects of relaxing assumptions about arrival rates and bandwidth.

Adaptive admission control in wireless multimedia networks under non-uniform traffic conditions

Adaptive admission control in cellular wireless networks reserves bandwidth for handoff calls, which is proportional to the traffic intensity in the surrounding cells. This paper analyzes performance (QoS) of the adaptive admission algorithm in the presence of nonuniform traffic conditions in the network. We model the performance of small, moderate, and large hot-spots, and analyze average cell capacity utilizations and their derivatives when the hot-spot new call arrival rate is growing and surrounding cells operate at the constant load. The analysis shows that under sufficient bandwidth reservation, derivatives of the hot-spot cell capacity utilizations converge to zero for hot-spot call arrival rates larger than twice the nominal load. Since the average utilization is the linear combination of all the state probabilities of the Markov chain, the convergence of its derivative to zero under high offered loads means that the derivatives of other linear combinations of state probabilities, such as handoff dropping probability and new call blocking probability, will also converge to zero. Therefore, handoff dropping probability and new call blocking probability must be bounded by the logarithmic function of new call arrival rate for large arrival rates. Due to the feedback property embedded in the bandwidth reservation process, our admission algorithm offers QoS bounds both under nominal load and under high arrival rates. The analytical results have been validated by simulations.

Dynamics and control of thin film growth process from a multicomponent gas

The control of a model system describing thin film growth processes from a multicomponent gas with chemical reactions occurring between components of source phase is studied. It is shown that condensation of thin films may be realized by different ways. In particular, for a low rate of particles arriving on a substrate the stable condensation mode occurs and any deviations from equilibrium are dumped. For a medium arrival rate, phase transitions take place in the oscillation mode (stable limit cycle). For a large arrival rate the stable limit cycle is destroyed and new phase condensation takes place in saw-tooth (accumulative) mode. A method of adaptive control of this oscillatory process - by varying the external parameters - is described based on recursive goal inequalities and an input - output linearized Poincaré map. The controlled system behaviour is analysed by means of computer simulations. It is found that the system being investigated has a peculiar memory because film condensation occurs in a different way for the same parameters but with differences in prehistory. It is noted that even small differences in the prehistory of the external parameters can lead to different modes of settling.

On the Transient Behavior of the Erlang Loss Model: Heavy Usage asymptotics

The Erlang loss model, which is the $M/M/m/m$ queue is considered. Asymptotic expansions are constructed for systems with a large number of servers $( m \gg 1 )$ and a large arrival rate that is $O( m )$. Formulas are given for the probability $p_n ( t )$ that n servers are occupied at time t. Several cases are treated of initial conditions and several regions in the $( {n,t} )$ plane. The approximations are constructed by using singular perturbation techniques such as the ray method, boundary layer theory, and the method of matched asymptotic expansions.

On the transient behavior of the m/m/m/m loss model

We develop an asymptotic technique for analyzing the transient behavior of queues and other stochastic models. As an example we consider the Erlang loss model which is the M/M/m/m queue. Asymptotic expansions are constructed for systems with a large number of servers and a large arrival rate that is O(m). We give formulas for the probability p n(t) that n servers are occupied at time t. We treat several cases of initial conditions and several regions in the (n, t) plane. Our approach is based on applying the ray method to a scaled form of the forward Kolmogorov equation which describes the time evolution of p n(t).

Distribution of the maximum buffer content during a busy period for state-dependent m/g/1 queues

We consider M/G/l queues characterized by the total unfinished work (buffer content) U(t) in the system at time t. We allow the rate of the Poisson arrivals as well as the rate at which the server works, to depend on the instantaneous value of U(t). In addition, the service density depends on the value of U(t) at the instant that the customer enters the system. We consider systems with a large arrival rate and small mean service times. We then construct asymptotic approximations to the probability that U(t) reaches the level K before completing the current busy period and we also compute the distribution of the maximum of U(t) during a busy period.