arXiv subjects

Sushil Mahavir Varma

Publications and source records attributed to Sushil Mahavir Varma.

16 recordsLinked to original sources

arXiv2026

Finite-Time Behavior of Erlang-C Model: Mixing Time, Mean Queue Length and Tail Bounds

Resource allocation problems in service systems like data centers and ride-hailing are usually studied using queueing models. Such systems are primarily studied in the steady-state and in asymptotic regimes such as under heavy traffic due to their analytical tractability. However, almost all applications in real life do not operate in asymptotic regimes, and so, there is a clear discrepancy in translating theoretical queuing results to practical applications. In this work, we bridge this gap by presenting nonasymptotic and finite-time bounds for Erlang-C systems, providing a stepping stone towards understanding the transient behavior of more general queuing systems. We bound the Chi-square distance between the finite-time queue length distribution and the stationary distribution, show that it decays exponentially fast, and characterize the rate of decay. We observe that the Erlang-C system exhibits a phase transition, depending on a parameter that measures the load relative to the size of the system. We then use these results to obtain bounds on the mean queue length and tails of the queue lengths in finite time for the nonasymptotic system. We also establish that the rate we obtain is tight up to universal constants in appropriate heavy-traffic asymptotic regimes. We obtain these results using the Lyapunov-Poincaré approach, where we first carefully design a Lyapunov function to obtain a negative drift outside a finite set. Within the finite set, we develop different strategies depending on the properties of the finite set to get a handle on the mixing behavior via a local Poincaré inequality. A key aspect of our methodological contribution is obtaining tight guarantees in these two regions, which when combined, give us tight mixing time bounds. We believe that this approach is of independent interest for studying mixing in reversible countable-state Markov chains more generally.

math.PR↗

arXiv2026

Efficiency-Reward Trade-Off in Queues with Dynamic Arrivals

Motivated by applications in online marketplaces such as ride-hailing platforms and payment channel networks, we study a single server queue with service rate $μ$ and queue-length-dependent arrival rate $λ(q) \in [0, λ_{\max}]$. The system operator receives a reward $F(λ)$ by setting an arrival rate $λ$, where $F$ can represent objectives such as throughput, revenue, or social welfare. The goal is to characterize the Pareto frontier between long-run average reward and expected queue length ($\mathbb{E}[q]$). We first establish a fluid upper bound on the reward achievable under any control policy $\{λ(q)\}$, and define regret as the gap from this upper bound. We then minimize $\mathbb{E}[q]$ subject to the regret being at most $\varepsilon$. In the small-market regime $(λ_{\max} \leq μ)$, we show that no control policy can improve over the classical heavy-traffic scaling of $\mathbb{E}[q] = Ω(1/\varepsilon)$. In contrast, in the large-market regime $(λ_{\max} > μ)$, we substantially improve over the classical scaling by explicitly constructing control policies with expected queue lengths at most $O(1/\sqrt{\varepsilon})$ or $O(\log (1/\varepsilon))$, depending on the curvature structure of $F$. Moreover, we establish universal lower bounds of $Ω(1/\sqrt{\varepsilon})$ or $Ω(\log (1/\varepsilon))$ respectively under any general control policy. These results can be viewed as (non asymptotic) heavy-traffic theory for queues with dynamic arrivals.

math.OC↗

arXiv2026

Transform Method for Stochastic Processing and Matching Networks

Modern service systems, ranging from cloud data centers and ride-hailing platforms to healthcare facilities, operate at massive scales where it is important to handle congestion. Queueing theory is used to understand the delay and queue length behavior in these systems. Except in simple queues, it is not possible to obtain a closed form solution for the quantities of interest, and so, one studies the system in certain asymptotic regimes such as the heavy traffic. The transform method, presented in this tutorial, is a framework to understand the steady-state behavior of Stochastic Processing and Matching Networks (SPNs/SMNs). By exploiting the zero-drift property of exponential test functions, the method derives explicit functional equations (acting as a proxy for global balance equations) for the transforms (such as moment-generating functions) of queue-length distributions. These functional equations can be used to either characterize the exact behavior of the system in an asymptotic regime or to obtain non-asymptotic performance bounds on the mean, higher order moments, or tail bounds on the queue lengths. Since its introduction for load-balancing in data center networks, the transform method, as a framework, has been extended to handle various features that arise in different systems, including customer abandonment, state-dependent arrivals, Markov-modulated arrivals, large-system scale, and multi-dimensional networks with multiple bottlenecks. This tutorial presents an overview of the transform method starting with the simplest setting viz., a single server queue. The transform method is introduced as a three step procedure. We then illustrate how the method can be adapted within this three-step framework to handle the aforementioned features.

math.OC↗

arXiv2026

Dynamic Pricing and Matching for Two-Sided Queues

Motivated by applications from gig economy and online marketplaces, we study a two-sided queueing system under joint pricing and matching controls. The queueing system is modeled by a bipartite graph, where the vertices represent customer or server types and the edges represent compatible customer-server pairs. Both customers and servers sequentially arrive to the system and join separate queues according to their types. The arrival rates of different types depend on the prices set by the system operator and the expected waiting time. At any point in time, the system operator can choose certain customers to match with compatible servers. The objective is to maximize the long-run average profit for the system. We first propose a fluid approximation based pricing and max-weight matching policy, which achieves an $O(\sqrtη)$ optimality rate when all the arrival rates are scaled by $η$. We further show that a two-price and max-weight matching policy achieves an improved $O(η^{1/3})$ optimality rate. Under a broad class of pricing policies, we prove that any matching policy has an optimality rate that is lower bounded by $Ω(η^{1/3})$. Thus, the latter policy achieves the optimal rate with respect to $η$. We also demonstrate the advantage of max-weight matching with respect to the number of server and customer types $n$. Under a complete resource pooling condition, we show that max-weight matching achieves $O(\sqrt{n})$ and $O(n^{1/3})$ optimality rates for static and two-price policies, respectively, and the latter matches the lower bound $Ω(n^{1/3})$. In comparison, the randomized matching policy may have an $Ω(n)$ optimality rate.

math.OC↗

arXiv2026

Online Stochastic Matching: A Polytope Perspective

Stochastic dynamic matching problems have recently gained attention in the stochastic-modeling community due to their diverse applications, such as supply-chain management and kidney exchange programs. In this paper, we study a matching problem where items of different classes arrive according to independent Poisson processes. Unmatched items are stored in a queue, and compatibility between items is represented by a simple graph, where items can be matched if their classes are connected. We analyze matching policies in terms of stability, delay, and long-term matching rate optimization. Our approach relies on the conservation equation, which ensures a balance between arrivals and departures in any stable system. Our main contributions are as follows. We establish a link between the existence of stable policies, the dimensionality of the solution set of the conservation equation, and the compatibility graph's structure. We describe the convex polytope formed by non-negative solutions to the conservation equation, and we design policies that can achieve or closely approximate the vertices of this polytope. When a vertex can only be approximated, we quantify the resulting trade-off between regret and delay: our policies achieve arbitrarily small regret at the cost of increasing delay, and we prove that this trade-off is unavoidable. Lastly, we discuss potential extensions of our results beyond the main assumptions of this paper.

cs.NI↗

arXiv2026

A Heavy Traffic Theory of Matching Queues

Motivated by emerging applications in online matching platforms and marketplaces, we study a matching queue. Customers and servers that arrive in a matching queue depart as soon as they are matched. While state-dependent control is an effective lever to regulate the throughput and delay, it often comes at a cost in practice for matching platforms. Optimizing this fundamental trade-off motivates the use of small amounts of control, so we study a matching queue in an asymptotic regime where the state-dependent control decreases to zero. Unlike the heavy traffic regime in classical queues, there are two different ways the control can be sent to zero, via a magnitude scaling parameter $ε$ that goes to zero and a time scaling parameter $τ$ that goes to infinity. Depending on the cost of control, we show that the rates of $ε$ and $τ$ that optimize the trade-off between delay and cost of control could correspond to three different regimes. As we traverse these regimes, we observe a phase transition in the limiting distribution of the matching queue. We show that a low cost of control corresponds to the regime $ετ\rightarrow 0$ and we call it the delay-driven regime. The limiting behavior in this regime is an asymmetrical Laplace distribution. On the other hand, $ετ\rightarrow \infty$ is the cost-driven regime corresponding to a high cost of control where the limiting behavior is either a uniform or a truncated exponential distribution. We christen the in-between regime of $ετ\rightarrow (0, \infty)$ the hybrid regime where the limiting behavior is a Gibbs distribution. These results are obtained by novel generalizations of the transform method, where each regime requires new ideas. The hybrid regime employs inverse Fourier transforms while the other two regimes engineer multiple complex exponential test functions.

math.PR↗

arXiv2026

Phase Transition in Convex Relaxations for Graph Alignment

We study the graph alignment problem for correlated Gaussian Orthogonal Ensemble (GOE) matrices, where the goal is to recover a hidden vertex permutation given two correlated symmetric Gaussian matrices $(A, B)$ with correlation $1/\sqrt{1+σ^2}$. While the maximum likelihood estimator is information-theoretically optimal, its computation, which reduces to a quadratic assignment problem, is intractable. Motivated by this, we analyze convex relaxations based on minimizing $\|AX - XB\|_F$ over the set of doubly stochastic matrices and the unit hypercube. We show that when the correlation parameter satisfies $σ= o(n^{-1/2}/\log^4 n)$, the solution of either relaxation $(X^\star)$ concentrates around the ground-truth permutation matrix $(Π^\star)$, i.e., $\|X^\star-Π^\star\|_F^2 = o(n)$, implying recovery of all but a vanishing fraction of vertices after simple post-processing. Combined with existing lower bounds, our results precisely characterize that $\|X^\star-Π^\star\|_F^2$ transitions from $o(n)$ for $σ= \tilde{o}(n^{-1/2})$ to $Ω(n)$ for $σ= \tildeΩ(n^{-1/2})$. In doing so, our analysis significantly tightens prior results and extends them beyond doubly stochastic relaxations.

stat.ML↗

arXiv2025

Near-Optimal Regret-Queue Length Tradeoff in Online Learning for Two-Sided Markets

We study a two-sided market, wherein, price-sensitive heterogeneous customers and servers arrive and join their respective queues. A compatible customer-server pair can then be matched by the platform, at which point, they leave the system. Our objective is to design pricing and matching algorithms that maximize the platform's profit, while maintaining reasonable queue lengths. As the demand and supply curves governing the price-dependent arrival rates may not be known in practice, we design a novel online-learning-based pricing policy and establish its near-optimality. In particular, we prove a tradeoff among three performance metrics: $\tilde{O}(T^{1-γ})$ regret, $\tilde{O}(T^{γ/2})$ average queue length, and $\tilde{O}(T^γ)$ maximum queue length for $γ\in (0, 1/6]$, significantly improving over existing results [1]. Moreover, barring the permissible range of $γ$, we show that this trade-off between regret and average queue length is optimal up to logarithmic factors under a class of policies, matching the optimal one as in [2] which assumes the demand and supply curves to be known. Our proposed policy has two noteworthy features: a dynamic component that optimizes the tradeoff between low regret and small queue lengths; and a probabilistic component that resolves the tension between obtaining useful samples for fast learning and maintaining small queue lengths.

cs.LG↗

arXiv2025

Graph Alignment via Birkhoff Relaxation

We consider the graph alignment problem, wherein the objective is to find a vertex correspondence between two graphs that maximizes the edge overlap. The graph alignment problem is an instance of the quadratic assignment problem (QAP), known to be NP-hard in the worst case even to approximately solve. In this paper, we analyze Birkhoff relaxation, a tight convex relaxation of QAP, and present theoretical guarantees on its performance when the inputs follow the Gaussian Wigner Model. More specifically, the weighted adjacency matrices are correlated Gaussian Orthogonal Ensemble with correlation $1/\sqrt{1+σ^2}$. Denote the optimal solutions of the QAP and Birkhoff relaxation by $Π^\star$ and $X^\star$ respectively. We show that $\|X^\star-Π^\star\|_F^2 = o(n)$ when $σ= o(n^{-1.25})$ and $\|X^\star-Π^\star\|_F^2 = Ω(n)$ when $σ= Ω(n^{-0.5})$. Thus, the optimal solution $X^\star$ transitions from a small perturbation of $Π^\star$ for small $σ$ to being well separated from $Π^\star$ as $σ$ becomes larger than $n^{-0.5}$. This result allows us to guarantee that simple rounding procedures on $X^\star$ align $1-o(1)$ fraction of vertices correctly whenever $σ= o(n^{-1.25})$. This condition on $σ$ to ensure the success of the Birkhoff relaxation is state-of-the-art.

stat.ML↗

arXiv2025

Electric Vehicle Fleet and Charging Infrastructure Planning

We study electric vehicle (EV) fleet and charging infrastructure planning in a spatial setting. With customer requests arriving continuously at rate $λ$ throughout the day, we determine the minimum number of vehicles and chargers for a target service level, along with matching and charging policies. While non-EV systems require extra $Θ(λ^{2/3})$ vehicles due to pickup times, EV systems differ. Charging increases nominal capacity, enabling pickup time reductions and allowing for an extra fleet requirement of only $Θ(λ^ν)$ for $ν\in (1/2, 2/3]$, depending on charging infrastructure and battery pack sizes. We propose the Power-of-$d$ dispatching policy, which achieves this performance by selecting the closest vehicle with the highest battery level from $d$ options. We extend our results to accommodate time-varying demand patterns and discuss conditions for transitioning between EV and non-EV capacity planning. Extensive simulations verify our scaling results, insights, and policy effectiveness while also showing the viability of low-range, low-cost fleets.

math.OC↗

arXiv2025

Power-of-$d$ Choices Load Balancing in the Sub-Halfin Whitt Regime

We consider the load balancing system under Poisson arrivals, exponential services, and homogeneous servers. Upon arrival, a job is to be routed to one of the servers, where it is queued until service. We consider the Power-of-$d$ choices routing algorithm, which chooses the queue with minimum length among $d$ randomly sampled queues. We study this system in the many-server heavy-traffic regime where the number of servers goes to infinity simultaneously when the load approaches the capacity. In particular, we consider a sequence of systems with $n$ servers and the arrival rate is given by $λ=n-n^{1-γ}$ for some $γ\in (0, 0.5)$, known as the sub-Halfin-Whitt regime. It was shown by [Liu Ying (2020)] that under Power-of-$d$ choices routing with $d \geq n^γ\log n$, the queue length behaves similarly to that of JSQ and that there are asymptotically zero queueing delays. The focus of this paper is to characterize the behavior when $d$ is below this threshold. We obtain high probability bounds on the queue lengths for various values of $d$ and large enough $n$. In particular, we show that when $d$ grows polynomially in $n$ but slower than in [Liu Ying (2020)], i.e., if $d$ is $Θ\left((n^γ\log n)^{1/m})\right)$ for some integer $m>1$, then the asymptotic queue length is $m$ with high probability. Moreover, if $d$ grows polylog in $n$, i.e., slower than any polynomial, but is at least $Ω(\log (n)^3)$, the queue length blows up to infinity asymptotically. We obtain these results by using an iterative state space collapse approach. We first establish a weak state-space collapse (SSC) on the queue lengths. Then, we bootstrap on weak SSC to iteratively narrow down the region of the collapse. After enough steps, this inductive refinement provides the bounds we seek. We establish these sequences of collapse using Lyapunov drift arguments.

math.PR↗

arXiv2023

Transportation Polytope and its Applications in Parallel Server Systems

A parallel server system is a stochastic processing network with applications in manufacturing, supply chain, ride-hailing, call centers, etc. Heterogeneous customers arrive in the system, and only a subset of servers can serve any customer type given by the flexibility graph. The goal of the system operator is to minimize the delay that depends on the scheduling policy and the flexibility graph. A long line of literature focuses on designing near-optimal scheduling policies given a flexibility graph. On the contrary, we fix the scheduling policy to be the so-called MaxWeight scheduling given its superior delay performance and focus on designing near-optimal, sparse flexibility graphs. Our contributions are threefold. First, we analyze the expected delay in the heavy-traffic asymptotic regime in terms of the properties of the flexibility graph and use this result to translate the design question in terms of transportation polytope, the deterministic equivalent of parallel server queues. Second, we design the sparsest flexibility graph that achieves a given delay performance and shows the robustness of the design to demand uncertainty. Third, given the budget to add edges arrives sequentially in time, we present the optimal schedule for adding them to the flexibility graph. These results are obtained by proving new results for transportation polytopes and are of independent interest. In particular, translating the difficulties to a simpler model, i.e. transportation polytope, allows us to develop a unified framework to answer several design questions.

cs.NI↗

arXiv2021

Throughput Optimal Routing in Blockchain Based Payment Systems

Cryptocurrency networks such as Bitcoin have emerged as a distributed alternative to traditional centralized financial transaction networks. However, there are major challenges in scaling up the throughput of such networks. Lightning network and Spider network are alternates that build bidirectional payment channels on top of cryptocurrency networks using smart contracts, to enable fast transactions that bypass the Blockchain. In this paper, we study the problem of routing transactions in such a payment processing network. We first propose a Stochastic model to study such a system, as opposed to a fluid model that is studied in the literature. Each link in such a model is a two-sided queue, and unlike classical queues, such queues are not stable unless there is an external control. We propose a notion of stability for the payment processing network consisting of such two-sided queues using the notion of on-chain rebalancing. We then characterize the capacity region and propose a throughput optimal algorithm that stabilizes the system under any load within the capacity region. The stochastic model enables us to study closed loop policies, which typically have better queuing/delay performance than the open loop policies (or static split rules) studied in the literature. We investigate this through simulations.

cs.DC↗

arXiv2021

Near Optimal Control in Ride Hailing Platforms with Strategic Servers

Motivated by applications in online marketplaces such as ride-hailing, we study how strategic servers impact the system performance. We consider a discrete-time process in which, heterogeneous types of customers and servers arrive. Each customer joins their type's queue, while servers might join a different type's queue depending on the prices posted by the system operator and an inconvenience cost. Then the system operator, constrained by a compatibility graph, decides the matching. The objective is to design an optimal control (pricing and matching scheme) to maximize the profit minus the expected waiting times. We develop a general framework that enables us to analyze a broad range of strategic behaviors. In particular, we encode servers' behavior in a properly defined \emph{cost function} that can be tailored to various settings. Using this general cost function, we introduce a novel probabilistic fluid problem. The probabilistic fluid model provides an upper bound on the achievable net profit. We then study the system under a large market regime in which the arrival rates are scaled by $η$ and present a probabilistic two-price policy and a max-weight matching policy which results in a net profit-loss of at most $O(η^{1/3})$. In addition, under a broad class of customer pricing policies, we show that any matching policy has net profit-loss of at least $Ω(η^{1/3})$. To show generality of our framework, we present multiple extensions to our model and analysis. We conclude the discussion by presenting numerical simulations comparing different cost models and analyzing performance of the proposed pricing and matching policies.

math.OC↗

arXiv2021

On the Linear convergence of Natural Policy Gradient Algorithm

Markov Decision Processes are classically solved using Value Iteration and Policy Iteration algorithms. Recent interest in Reinforcement Learning has motivated the study of methods inspired by optimization, such as gradient ascent. Among these, a popular algorithm is the Natural Policy Gradient, which is a mirror descent variant for MDPs. This algorithm forms the basis of several popular Reinforcement Learning algorithms such as Natural actor-critic, TRPO, PPO, etc, and so is being studied with growing interest. It has been shown that Natural Policy Gradient with constant step size converges with a sublinear rate of O(1/k) to the global optimal. In this paper, we present improved finite time convergence bounds, and show that this algorithm has geometric (also known as linear) asymptotic convergence rate. We further improve this convergence result by introducing a variant of Natural Policy Gradient with adaptive step sizes. Finally, we compare different variants of policy gradient methods experimentally.

cs.LG↗

arXiv2020

Logarithmic Heavy Traffic Error Bounds in Generalized Switch and Load Balancing Systems

Motivated by application in wireless networks, cloud computing, data centers etc, Stochastic Processing Networks have been studied in the literature under various asymptotic regimes. In the heavy-traffic regime, the steady state mean queue length is proved to be $O(\frac{1}ε)$ where $ε$ is the heavy-traffic parameter, that goes to zero in the limit. The focus of this paper is on obtaining queue length bounds on prelimit systems, thus establishing the rate of convergence to the heavy traffic. In particular, we study the generalized switch model operating under the MaxWeight algorithm, and we show that the mean queue length of the prelimit system is only $O\left(\log \left(\frac{1}ε\right)\right)$ away from its heavy-traffic limit. We do this even when the so called complete resource pooling (CRP) condition is not satisfied. When the CRP condition is satisfied, in addition, we show that the MaxWeight algorithm is within $O\left(\log \left(\frac{1}ε\right)\right)$ of the optimal. Finally, we obtain similar results in load balancing systems operating under the join the shortest queue routing algorithm.

math.PR↗