arXiv subjects

Seva Shneer

Publications and source records attributed to Seva Shneer.

30 records · Page 2Linked to original sources

arXiv2020

Static vs accumulating priorities in healthcare queues under heavy loads

Amid unprecedented times caused by COVID-19, healthcare systems all over the world are strained to the limits of, or even beyond, capacity. A similar event is experienced by some healthcare systems regularly, due to for instance seasonal spikes in the number of patients. We model this as a queueing system in heavy traffic (where the arrival rate is approaching the service rate from below) or in overload (where the arrival rate exceeds the service rate). In both cases we assume that customers (patients) may have different priorities and we consider two popular service disciplines: static priorities and accumulating priorities. It has been shown that the latter allows for patients of all classes to be seen in a timely manner as long as the system is stable. We demonstrate however that if accumulating priorities are used in the heavy traffic or overload regime, then all patients, including those with the highest priority, will experience very long waiting times. If on the other hand static priorities are applied, then one can ensure that the highest-priority patients will be seen in a timely manner even in overloaded systems.

cs.PF↗

arXiv2020

Stability of JSQ in queues with general server-job class compatibilities

We consider Poisson streams of exponentially distributed jobs arriving at each edge of a hypergraph of queues. Upon arrival, an incoming job is rooted to the shortest queue among the corresponding vertices. This generalizes many known models such as power-of-d load balancing and JSQ (join the shortest queue) on generic graphs. We provide a generic condition for stability of this model. We show that some graph topologies lead to a loss of capacity, implying more restrictive stability conditions than in, e.g., complete graphs.

math.PR↗

arXiv2019

MDS coding is better than replication for job completion times

In a multi-server system, how can one get better performance than random assignment of jobs to servers if queue-states cannot be queried by the dispatcher? A replication strategy has recently been proposed where $d$ copies of each arriving job are sent to servers chosen at random. The job's completion time is the first time that the service of any of its copies is complete. On completion, redundant copies of the job are removed from other queues so as not to overburden the system. For digital jobs, where the objects to be served can be algebraically manipulated, and for servers whose output is a linear function of their input, here we consider an alternate strategy: Maximum Distance Separable (MDS) codes. For every batch of $n$ digital jobs that arrive, $n+m$ linear combinations are created over the reals or a large finite field, and each coded job is sent to a random server. The batch completion time is the first time that any $n$ of the $n+m$ coded jobs are served, as the evaluation of $n$ original jobs can be recovered by Gaussian elimination. If redundant jobs can be removed from queues on batch completion, we establish that in order to get the improved response-time performance of sending $d$ copies of each of $n$ jobs via the replication strategy, with the MDS methodology it suffices to send $n+d$ jobs. That is, while replication is multiplicative, MDS is linear.

cs.PF↗

arXiv2019

Stability and moment bounds under utility-maximising service allocations: finite and infinite networks

We study networks of interacting queues governed by utility-maximising service-rate allocations in both discrete and continuous time. For {\em finite} networks we establish stability and some steady-state moment bounds under natural conditions and rather weak assumptions on utility functions. These results are obtained using direct applications of Lyapunov-Foster-type criteria, and apply to a wide class of systems, including those for which fluid limit-based approaches are not applicable. We then establish stability and some steady-state moment bounds for two classes of {\em infinite} networks, with single-hop and multi-hop message routes. These results are proved by considering the infinite systems as limits of their truncated finite versions. The uniform moment bounds for the finite networks play a key role in these limit transitions.

math.PR↗

arXiv2019

Discrete-time TASEP with holdback

We study the following interacting particle system. There are $ρn$ particles, $ρ< 1$, moving clockwise ("right"), in discrete time, on $n$ sites arranged in a circle. Each site may contain at most one particle. At each time, a particle may move to the right-neighbor site according to the following rules. If its right-neighbor site is occupied by another particle, the particle does not move. If the particle has unoccupied sites ("holes") as neighbors on both sides, it moves right with probability $1$. If the particle has a hole as the right-neighbor and an occupied site as the left-neighbor, it moves right with probability $0 h$, a {\em condensation} phenomenon occurs, namely the formation and persistence of large particle clusters; in particular, the typical flux in this case is $p(1-ρ) < h < ρ$, which differs from the formal flux when $h < ρ< 1/2$. Our results include both steady-state and transient analysis. In particular, we derive a version of the Ballot Theorem, and show that the key "reason" for large cluster formation for densities $ρ> h$ is described by this theorem.

math.PR↗

arXiv2019

Stability conditions for a decentralised medium access algorithm: single- and multi-hop networks

We consider a decentralised multi-access algorithm, motivated primarily by the control of transmissions in a wireless network. For a finite single-hop network with arbitrary interference constraints we prove stochastic stability under the natural conditions. For infinite and finite single-hop networks, we obtain broad rate-stability conditions. We also consider symmetric finite multi-hop networks and show that the natural condition is sufficient for stochastic stability.

cs.IT↗

arXiv2019

The end time of SIS epidemics driven by random walks on edge-transitive graphs

Network epidemics is a ubiquitous model that can represent different phenomena and finds applications in various domains. Among its various characteristics, a fundamental question concerns the time when an epidemic stops propagating. We investigate this characteristic on a SIS epidemic induced by agents that move according to independent continuous time random walks on a finite graph: Agents can either be infected (I) or susceptible (S), and infection occurs when two agents with different epidemic states meet in a node. After a random recovery time, an infected agent returns to state S and can be infected again. The End of Epidemic (EoE) denotes the first time where all agents are in state S, since after this moment no further infections can occur and the epidemic stops. For the case of two agents on edge-transitive graphs, we characterize EoE as a function of the network structure by relating the Laplace transform of EoE to the Laplace transform of the meeting time of two random walks. Interestingly, this analysis shows a separation between the effect of network structure and epidemic dynamics. We then study the asymptotic behavior of EoE (asymptotically in the size of the graph) under different parameter scalings, identifying regimes where EoE converges in distribution to a proper random variable or to infinity. We also highlight the impact of different graph structures on EoE, characterizing it under complete graphs, complete bipartite graphs, and rings.

math.PR↗

arXiv2019

The variance of the average depth of a pure birth process converges to 7

If trees are constructed from a pure birth process and one defines the depth of a leaf to be the number of edges to its root, it is known that the variance in the depth of a randomly selected leaf of a randomly selected tree grows linearly in time. In this letter, we instead consider the variance of the average depth of leaves within each individual tree, establishing that, in contrast, it converges to a constant, $7$. This result indicates that while the variance in leaf depths amongst the ensemble of pure birth processes undergoes large fluctuations, the average depth across individual trees is much more consistent.

math.PR↗

arXiv2018

Counting cliques and cycles in scale-free inhomogeneous random graphs

Scale-free networks contain many small cliques and cycles. We model such networks as inhomogeneous random graphs with regularly varying infinite-variance weights. For these models, the number of cliques and cycles have exact integral expressions amenable to asymptotic analysis. We obtain various asymptotic descriptions for how the average number of cliques and cycles, of any size, grow with the network size. For the cycle asymptotics we invoke the theory of circulant matrices.

math.PR↗

arXiv2018

Stability conditions for a discrete-time decentralised medium access algorithm

We consider a stochastic queueing system modelling the behaviour of a wireless network with nodes employing a discrete-time version of the standard decentralised medium access algorithm. The system is {\em unsaturated} -- each node receives an exogenous flow of packets at the rate $λ$ packets per time slot. Each packet takes one slot to transmit, but neighboring nodes cannot transmit simultaneously. The algorithm we study is {\em standard} in that: a node with empty queue does {\em not} compete for medium access; the access procedure by a node does {\em not} depend on its queue length, as long as it is non-zero. Two system topologies are considered, with nodes arranged in a circle and in a line. We prove that, for either topology, the system is stochastically stable under condition $λ< 2/5$. This result is intuitive for the circle topology as the throughput each node receives in a saturated system (with infinite queues) is equal to the so called {\em parking constant}, which is larger than $2/5$. (The latter fact, however, does not help to prove our result.) The result is not intuitive at all for the line topology as in a saturated system some nodes receive a throughput lower than $2/5$.

cs.NI↗

arXiv2012

Stability of a Markov-modulated Markov Chain, with application to a wireless network governed by two protocols

We consider a discrete-time Markov chain $(X^t,Y^t)$, $t=0,1,2,...$, where the $X$-component forms a Markov chain itself. Assume that $(X^t)$ is Harris-ergodic and consider an auxiliary Markov chain ${\hat{Y}^t}$ whose transition probabilities are the averages of transition probabilities of the $Y$-component of the $(X,Y)$-chain, where the averaging is weighted by the stationary distribution of the $X$-component. We first provide natural conditions in terms of test functions ensuring that the $\hat{Y}$-chain is positive recurrent and then prove that these conditions are also sufficient for positive recurrence of the original chain $(X^t,Y^t)$. The we prove a "multi-dimensional" extension of the result obtained. In the second part of the paper, we apply our results to two versions of a multi-access wireless model governed by two randomised protocols.

math.PR↗

arXiv2009

Heavy-traffic analysis of the maximum of an asymptotically stable random walk

For families of random walks $\{S_k^{(a)}\}$ with $\mathbf E S_k^{(a)} = -ka < 0$ we consider their maxima $M^{(a)} = \sup_{k \ge 0} S_k^{(a)}$. We investigate the asymptotic behaviour of $M^{(a)}$ as $a \to 0$ for asymptotically stable random walks. This problem appeared first in the 1960's in the analysis of a single-server queue when the traffic load tends to 1 and since then is referred to as the heavy-traffic approximation problem. Kingman and Prokhorov suggested two different approaches which were later followed by many authors. We give two elementary proofs of our main result, using each of these approaches. It turns out that the main technical difficulties in both proofs are rather similar and may be resolved via a generalisation of the Kolmogorov inequality to the case of an infinite variance. Such a generalisation is also obtained in this note.

math.PR↗