arXiv subjects

Jamol Pender

Publications and source records attributed to Jamol Pender.

At least 19 recordsLinked to original sources

arXiv2026

Erlang Loss Model with Energy Constrained Servers

In this paper, we study an Erlang-type loss system with energy-constrained servers. Each server is equipped with a finite battery and becomes temporarily unavailable for service when its energy is depleted, entering a charging phase before returning to operation upon full recharge. Customers who arrive to find all servers unavailable are blocked and lost immediately. We characterize the steady-state behavior of this two-dimensional Markov process by extending classical truncation results for Jackson networks to incorporate energy dynamics. This yields a closed-form product-form stationary distribution, from which we derive explicit expressions for the steady-state moments and the blocking probability. Finally, we establish that the corresponding M/G/$k$/$k$ queue with stochastic charging is insensitive to both the service-time and charging-time distributions, depending only on their means. Thus, we extend the insensitivity property of the Erlang loss queue to the stochastic server setting.

math.PR↗

arXiv2026

The Amplitude Dynamics of Impulsive Queues

In this paper, we analyze a multiserver Markovian queue with customer abandonment i.e. the Erlang-A queue uder a novel framework, i.e. the random impulsive differential equations (RIDEs). This framework captures systems that evolve continuously while experiencing sudden, discrete interventions. The combination of such framework with Erlang-A queue give rise to multiple real-life applications, such as improving efficiency at call centers and designing optimal timing to apply quantum error correction in trying to shun away from decoherence in quantum computing. We derive closed-form expressions for steady-state amplitude bounds and average queue lengths under impulse, and we identify the impulse timings that optimize system performance.

math.PR↗

arXiv2026

Telehealth Control Policies: Bridging the Gap Between Patients and Doctors

This paper studies a sequential decision-making problem in a two-stage queueing system modeled after operations in CVS MinuteClinics, where nurse practitioners (NPs) oversee patient care throughout the entire visit. All services are non-preemptive, and NPs cannot begin treating a new patient until the current patient has completed both stages of care. Following an initial diagnosis in the upstream phase, NPs must decide for low-acuity patients whether to proceed with treatment independently through immediate service, or to collaborate with a dedicated general physician (GP) via telemedicine. While collaboration typically improves service quality and is preferred by individual patients, it may introduce delays as the NP-patient pair waits for a GP to become available. This work explores the structural properties of optimal policies under different system parameters, with a focus on large initial upstream queues, revealing unconventional and complex policy behaviors. Leveraging these structural insights and supporting theoretical results, we design simple and effective heuristics that are computable in linear time and suitable for practical implementation. These heuristics are robust across the entire parameter space of interest, and offer clear, actionable guidance for NPs as system parameters vary. They also achieve near-optimal performance, averaging within 0.1% of the optimal, while commonly used benchmark policies are highly sensitive to parameter shifts and can incur costs more than 100% higher than optimal. The work provides applicable insights for decision-makers on improving policy robustness and effectiveness, as well as recommendations for stakeholders on the value of investing in telemedicine infrastructure. For instance, we identify scenarios where such investments may be either unnecessary or essential based on specific system parameters.

math.OC↗

arXiv2026

Queues with Rechargeable Servers

Drone delivery systems violate a core assumption in classical queueing models: server capacity is not fixed. Drones (servers) periodically must recharge, creating random fluctuations in service availability. We introduce an Erlang--S$^{*}$ queue that incorporates charging dynamics (probability of charging after service completion $p$ and charging return rate $\gamma$) together with abandonment. We derive fluid and diffusion limits, yielding closed-form steady-state means, variances, and covariances for the joint queue--server process $(Q,S)$. The diffusion limits allow us to derive new staffing rules for the probability of delay and the probability of abandonment targets. A key insight is that server stochasticity induces systematic capacity loss relative to fixed--server systems, leading to a regime--dependent staffing adjustment: additive shifts in underloaded regimes and multiplicative scaling in overloaded regimes. Our simulation experiments confirm both the accuracy of the limit theorems and the performance of the staffing schedule's ability to achieve their targets.

math.PR↗

arXiv2026

Control policies for a two-stage queueing system with parallel and single server options

We study a two-stage tandem service queue attended by two servers. Each job-server pair must complete both service phases together, with the server unable to begin a new job until the current one is fully processed after two stages. Immediately after the first phase of service, the server decides whether to send the job/customer to a downstream station that allows parallel processing or to a single-service facility that offers faster or higher-quality service but handles only one job at a time. This choice determines whether the second phase commences immediately or (potentially) after waiting in a queue for the single-service facility to become available. The decision-making scenario is modeled via a Markov decision process formulation, of a clearing system with holding costs at each station. We fully characterize the structural properties of an optimal control policy based on the relationship between the service rates at the downstream stations. A numerical study highlights the significance of optimal control by comparing its performance against several natural heuristic policies.

math.OC↗

arXiv2026

Balancing Independent and Collaborative Service

We study a two-type server queueing system where flexible Type-I servers, upon their initial interaction with jobs, decide in real time whether to process them independently or in collaboration with dedicated Type-II servers. Independent processing begins immediately, as does collaborative service if a Type-II server is available. Otherwise, the job and its paired Type-I server wait in queue for collaboration. Type-I servers are non-preemptive and cannot engage with new jobs until their current job is completed. We provide a complete characterization of the structural properties of the optimal policy for the clearing system. In particular, an optimal control is shown to follow a threshold structure based on the number of jobs in the queue before a Type-I first interaction and on the number of jobs in either independent or collaborative service. We propose simple threshold heuristics, based on linear approximations, for real-time decision-making. In much of the parameter and state spaces, we establish theoretical bounds that compare the thresholds proposed by our heuristics to those of optimal policies and identify parameter configurations where these bounds are attained. Outside of these regions, the optimal thresholds are infinite. Numerical experiments further demonstrate the accuracy and robustness of our heuristics, particularly when the initial queue length is high. Our proposed heuristics achieve costs within 0.5% of the optimal policy on average and significantly outperform benchmark policies that exhibit extreme sensitivity to system parameters, sometimes incurring costs exceeding 100% of the optimal.

math.OC↗

arXiv2025

Community Bail Fund Systems: Fluid Limits and Approximations

Community bail funds (CBFs) assist individuals who have been arrested and cannot afford bail, preventing unnecessary pretrial incarceration along with its harmful or sometimes fatal consequences. By posting bail, CBFs allow defendants to stay at home and maintain their livelihoods until trial. This paper introduces new stochastic models that combine queueing theory with classic insurance risk models to capture the dynamics of the remaining funds in a CBF. We first analyze a model where all bail requests are accepted. Although the remaining fund balance can go negative, this model provides insight for CBFs that are not financially constrained. We then apply the Skorokhod map to make sure the CBF balance does not go negative and show that the Skorokhod map produces a model where requests are partially fulfilled. Finally, we analyze a model where bail requests can be blocked if there is not enough money to satisfy the request upon arrival. Although the blocking model prevents the CBF from being negative, the blocking feature gives rise to new analytical challenges for a direct stochastic analysis. Thus, we prove a functional law of large numbers or a fluid limit for the blocking model and show that the fluid limit is a distributed delay equation. We assess the quality of our fluid limit via simulation and show that the fluid limit accurately describes the large-scale stochastic dynamics of the CBF. Finally, we prove stochastic ordering results for the CBF processes we analyze.

math.PR↗

arXiv2023

The Maximum Overlap Time in the M/M/1 Queue

In this paper, we analyze the steady state maximum overlap time in the M/M/1 queue. We derive the maximum overlap time tail distribution, its moments and the moment generating function. We also analyze the steady state minimum overlap time of the adjacent customers and compute its moments and moment generating function. Our results provide new insight on how customers become infected in the M/M/1 queue.

math.PR↗

arXiv2023

Overlap Times in Tandem Queues: Identically Distributed Station Case

In this paper, we investigate overlap times in a two-dimensional infinite server tandem queue. Specifically, we analyze the amount of time that a pair of customers spend overlapping in any station of the two dimensional tandem network. We assume that both stations have independent and identically distributed exponential service times with the same rate parameter $\mu$. Our main contribution is the derivation of the joint tail distribution, the two marginal tail probabilities, the moments of the overlap times and the tail distribution of the sum of the overlap times in both stations. Our results shed light on how customers overlap downstream in serial queueing systems.

math.PR↗

arXiv2023

The Number of Overlapping Customers in Erlang-A Queues: An Asymptotic Approach

In this paper, we investigate the number of customers that overlap or coincide with a virtual customer in an Erlang-A queue. Our study provides a novel approach that exploits fluid and diffusion limits for the queue to approximate the mean and variance of the number of overlapping customers. We conduct a detailed analysis of the fluid and diffusion limit differential equations to derive these approximations. We also construct new accurate approximations for the mean and variance of the waiting time in the Erlang-A queue by combining fluid limits with the polygamma function. Our findings have important implications for queueing theory and evaluating the overlap risk of more complicated service systems.

math.PR↗

arXiv2023

Overlap Times in the $GI^B/GI/\infty$ Queue

Overlap times have been studied as a way of understanding the time of interaction between customers in a service facility. Most of the previous analysis relies on the single jump assumption for arrivals, which implies the queue increases by one for each arrival epoch. In this paper, we relax the single arrival assumption and explore the impact of having batch arrivals. Unfortunately, with batch arrivals it is not clear how one measures an overlap time between batches of customers. Thus, we develop two ways of capturing the notion of an overlap time in a batch setting and derive exact results in the infinite server queue with batch arrivals. Finally, we derive new results for analyzing overlap times of more than two batches.

math.PR↗

arXiv2022

Queues with Delayed Information: Analyzing the Impact of the Choice Model Function

In this paper, we study queueing systems with delayed information that use a generalization of the multinomial logit choice model as its arrival process. Previous literature assumes that the functional form of the multinomial logit model is exponential. However, in this work we generalize this to different functional forms. In particular, we compute the critical delay and analyze how it depends on the choice of the functional form. We highlight how the functional form of the model can be interpreted as an exponential model where the exponential rate parameter is uncertain. Furthermore, the rate parameter distribution is given by the inverse Laplace-Stieltjes transform of the functional form when it exists. We perform numerous numerical experiments to confirm our theoretical insights.

math.DS↗

arXiv2021

Mean Field Queues with Delayed Information

In this paper, we consider a new queueing model where queues balance themselves according to a mean field interaction with a time delay. Unlike other work with delayed information our model considers multi-server queues with customer abandonment. In this setting, our queueing model corresponds to a system of mean field interacting delay differential equations with a point of non-differentiability introduced by the finite-server and abandonment terms. We show that this system of delay differential equations exhibits a change in stability when the delay in information crosses a critical threshold. In particular, the system exhibits periodic oscillations when the delay in information exceeds this critical threshold and we show that the threshold surprisingly does not depend on the number of queues. This is in stark contrast to other choice based queueing models with delayed information. We compute this critical threshold in each of the relevant parameter regions induced by the point of non-differentiability and show numerically how the critical threshold transitions through the point of non-differentiability.

math.DS↗

arXiv2021

A Note on the Interpretation of Distributed Delay Equations

Distributed delay equations have been used to model situations in which there is some sort of delay whose duration is uncertain. However, the interpretation of a distributed delay equation is actually very different from that of a delay differential equation with a random delay. This work explicitly highlights this distinction as it is an important consideration to make when modeling delayed systems in which the delay can take on several values.

math.DS↗

arXiv2021

Overlap Times in the Infinite Server Queue

Imagine, you enter a grocery store to buy food. How many peopledo you overlap with in this store? How much time do you overlap witheach person in the store? In this paper, we answer these questions bystudying the overlap times between customers in the infinite serverqueue. We compute in closed form the steady state distribution ofthe overlap time between a pair of customers and the distribution ofthe number of customers that an arriving customer will overlap with.Finally, we define a residual process that counts the number of over-lapping customers that overlap in the queue for at least{\delta}time unitsand compute its mean, variance, and distribution in the exponentialservice setting

math.PR↗

arXiv2021

Queues with Updating Information: Finding the Amplitude of Oscillations

Many service systems provide customers with information about the system so that customers can make an informed decision about whether to join or not. Many of these systems provide information in the form of an update. Thus, the information about the system is updated periodically in increments of size $\Delta$. It is known that these updates can cause oscillations in the resulting dynamics. However, it is an open problem to explicitly characterize the size of these oscillations when they occur. In this paper, we solve this open problem and show how to exactly calculate the amplitude of these oscillations via a fixed point equation. We also calculate closed form approximations via Taylor expansions of the fixed point equation and show that these approximations are very accurate, especially when $\Delta$ is large. Our analysis provides new insight for systems that use updates as a way of disseminating information to customers.

math.DS↗

arXiv2021

An Ephemerally Self-Exciting Point Process

Across a wide variety of applications, the self-exciting Hawkes process has been used to model phenomena in which the history of events influences future occurrences. However, there may be many situations in which the past events only influence the future as long as they remain active. For example, a person spreads a contagious disease only as long as they are contagious. In this paper, we define a novel generalization of the Hawkes process that we call the ephemerally self-exciting process. In this new stochastic process, the excitement from one arrival lasts for a randomly drawn activity duration, hence the ephemerality. Our study includes exploration of the process itself as well as connections to well-known stochastic models such as branching processes, random walks, epidemics, preferential attachment, and Bayesian mixture models. Furthermore, we prove a batch scaling construction of general, marked Hawkes processes from a general ephemerally self-exciting model, and this novel limit theorem both provides insight into the Hawkes process and motivates the model contained herein as an attractive self-exciting process in its own right.

math.PR↗

arXiv2020

Multi-Delay Differential Equations: A Taylor Expansion Approach

It is already well-understood that many delay differential equations with only a single constant delay exhibit a change in stability according to the value of the delay in relation to a critical delay value. Finding a formula for the critical delay is important to understanding the dynamics of delayed systems and is often simple to obtain when the system only has a single constant delay. However, if we consider a system with multiple constant delays, there is no known way to obtain such a formula that determines for what values of the delays a change in stability occurs. In this paper, we present some single-delay approximations to a multi-delay system obtained via a Taylor expansion as well as formulas for their critical delays which are used to approximate where the change in stability occurs in the multi-delay system. We determine when our approximations perform well and we give extra analytical and numerical attention to the two-delay and three-delay settings.

math.DS↗