Jamol Pender

The truncated normal distribution

Motivated by heavy traffic approximations for single server queues with abandonment, we provide an exact expression for the moments of the truncated normal distribution using Steins lemma. Consequently, our moment expressions provide insight into the steady state skewness and kurtosis dynamics of single server queues with impatient customers. Moreover, based on the truncated normal distribution, we develop a new approximation for single server queues with abandonment in the nonstationary setting. Numerical examples illustrate that our approximation performs quite well.

NONSTATIONARY LOSS QUEUES VIA CUMULANT MOMENT APPROXIMATIONS

In this paper, we provide a new technique for analyzing the nonstationary Erlang loss queueing model with abandonment. Our method uniquely combines the use of the functional Kolmogorov forward equations with the well-known Gram-Charlier series expansion from the statistics literature. Using the Gram-Charlier series expansion, we show that we can estimate salient performance measures of the loss queue such as the mean, variance, skewness, kurtosis, and blocking probability. Lastly, we provide numerical examples to illustrate the effectiveness of our approximations.

A Poisson–Charlier approximation for nonstationary queues

Abstract In this paper, we develop a new approximation for nonstationary multiserver queues with abandonment. Our method uses the Poisson–Charlier polynomials, which are a discrete orthogonal polynomial sequence that is orthogonal with respect to the Poisson distribution. We show that by appealing to the Poisson–Charlier polynomials that we can estimate the mean, variance, and probability of delay of our nonstationary queueing system with good accuracy. Lastly, we provide a numerical example that illustrates that our approximations are effective.

An analysis of nonstationary coupled queues

We consider a two dimensional time varying tandem queue with coupled processors. We assume that jobs arrive to the first station as a non-homogeneous Poisson process. When each queue is non-empty, jobs are processed separately like an ordinary tandem queue. However, if one of the processors is empty, then the total service capacity is given to the other processor. This problem has been analyzed in the constant rate case by leveraging Riemann Hilbert theory and two dimensional generating functions. Since we are considering time varying arrival rates, generating functions cannot be used as easily. Thus, we choose to exploit the functional Kolmogorov forward equations (FKFE) for the two dimensional queueing process. In order to leverage the FKFE, it is necessary to approximate the queueing distribution in order to compute the relevant expectations and covariance terms. To this end, we expand our two dimensional Markovian queueing process in terms of a two dimensional polynomial chaos expansion using the Hermite polynomials as basis elements. Truncating the polynomial chaos expansion at a finite order induces an approximate distribution that is close to the original stochastic process. Using this truncated expansion as a surrogate distribution, we can accurately estimate probabilistic quantities of the two dimensional queueing process such as the mean, variance, and probability that each queue is empty.

Sampling the Functional Kolmogorov Forward Equations for Nonstationary Queueing Networks

Nonstationary queueing networks are often difficult to approximate. Recent novel methods for approximating the moments of nonstationary queues use the functional version of the Kolmogorov forward equations in conjunction with orthogonal polynomial expansions. However, these methods require closed form expressions for the expectations that appear in the functional Kolmogorov forward equations. When closed form expressions cannot be easily derived, these methods cannot be used. In this paper, we present a new sampling algorithm to overcome this difficulty; our sampling algorithm accurately estimates the expectations using simulation. We apply our algorithm to priority queues, which are useful for modeling hospital triage systems. We show that our sampling algorithm accurately estimates the mean and variance of the priority queue without spending significantly more computational time than integrating ordinary differential equations. Last, we compare our sampling algorithm to the closed form analytical approx...

Approximations for the Queue Length Distributions of Time-Varying Many-Server Queues

This paper presents a novel and computationally efficient methodology for approximating the queue length (the number of customers in the system) distributions of time-varying non-Markovian many-server queues (e.g., Gt/Gt/nt queues), where the number of servers (nt) is large. Our methodology consists of two steps. The first step uses phase-type distributions to approximate the general interarrival and service times, thus generating an approximating Pht/Pht/nt queue. The second step develops strong approximation theory to approximate the Pht/Pht/nt queue with fluid and diffusion limits whose mean and variance can be computed using ordinary differential equations. However, by naively representing the Pht/Pht/nt queue as a Markov process by expanding the state space, we encounter the lingering phenomenoneven when the queue is overloaded. Lingering typically occurs when the mean queue length is equal or near the number of servers, however, in this case it also happens when the queue is overloaded and this time...

Comparisons of ticket and standard queues

Upon arrival to a ticket queue, a customer is offered a slip of paper with a number on it—indicating the order of arrival to the system—and is told the number of the customer currently in service. The arriving customer then chooses whether to take the slip or balk, a decision based on the perceived queue length and associated waiting time. Even after taking a ticket, a customer may abandon the queue, an event that will be unobservable until the abandoning customer would have begun service. In contrast, a standard queue has a physical waiting area so that abandonment is apparent immediately when it takes place and balking is based on the actual queue length at the time of arrival. We prove heavy traffic limit theorems for the generalized ticket and standard queueing processes, discovering that the processes converge together to the same limit, a regulated Ornstein–Uhlenbeck process. One conclusion is that for a highly utilized service system with a relatively patient customer population, the ticket and standard queue performances are asymptotically indistinguishable on the scale typically uncovered under heavy traffic approaches. Next, we heuristically estimate several performance metrics of the ticket queue, some of which are of a sensitivity typically undetectable under diffusion scaling. The estimates are tested using simulation and are shown to be quite accurate under a general collection of parameter settings.

Diffusion limits for the (MAPt/Pht/)N queueing network

In this paper, we prove strong approximations for the (MAPt/Pht/)N queueing network. These strong approximations allow us to derive fluid and diffusion limits for the queue length processes of the network. This extends recent work that provides fluid and diffusion limits in the single station setting.

Strong approximations for time-varying infinite-server queues with non-renewal arrival and service processes

ABSTRACT In real stochastic systems, the arrival and service processes may not be renewal processes. For example, in many telecommunication systems such as internet traffic where data traffic is bursty, the sequence of inter-arrival times and service times are often correlated and dependent. One way to model this non-renewal behavior is to use Markovian Arrival Processes (MAPs) and Markovian Service Processes (MSPs). MAPs and MSPs allow for inter-arrival and service times to be dependent, while providing the analytical tractability of simple Markov processes. To this end, we prove fluid and diffusion limits for MAPt/MSPt/∞ queues by constructing a new Poisson process representation for the queueing dynamics and leveraging strong approximations for Poisson processes. As a result, the fluid and diffusion limit theorems illuminate how the dependence structure of the arrival or service processes can affect the sample path behavior of the queueing process. Finally, our Poisson representation for MAPs and MSPs is useful for simulation purposes and may be of independent interest.

A Law of Large Numbers for M/M/c/Delayoff-Setup Queues with Nonstationary Arrivals

Cloud computing is a new paradigm where a company makes money by selling computing resources including both software and hardware. The core part or infrastructure of cloud computing is the data center where a large number of servers are available for processing incoming data traffic. These servers not only consume a large amount of energy to process data, but also need a large amount of energy to keep cool. Therefore, a reduction of a few percent of the power consumption means saving a substantial amount of money for the company as well as reduce our impact on the environment. As it currently stands, an idle server still consumes about 60 % of its peak energy usage. Thus, a natural suggestion to reduce energy consumption is to turn off servers which are not processing data. However, turning off servers can affect the customer experience. Customers trying to access computing power will experience delays if their data cannot be processed quickly enough. Moreover, servers require setup times in order to move from the off state to the on state. In the setup phase, servers consume energy, but cannot process data. Therefore, there exists a trade-off between power consumption and delay performance. In [7, 9], the authors analyze this tradeoff using an M/M/c queue with setup time for which they present a decomposition property by solving difference equations. In this paper, we complement recent stationary analysis of these types of models by studying the sample path behavior of the queueing model. In this regard, we prove a weak law of large numbers or fluid limit theorem for the queue length and server processes as the number of arrivals and number of servers tends to infinity. This methodology allows us to consider the impact of nonstationary arrivals and abandonment, which have not been considered in the literature so far.