Lindley's Recursion Research Articles

Sample-Path Large Deviations for Unbounded Additive Functionals of the Reflected Random Walk

We prove a sample-path large deviation principle (LDP) with sublinear speed for unbounded functionals of certain Markov chains induced by the Lindley recursion. The LDP holds in the Skorokhod space [Formula: see text] equipped with the [Formula: see text] topology. Our technique hinges on a suitable decomposition of the Markov chain in terms of regeneration cycles. Each regeneration cycle denotes the area accumulated during the busy period of the reflected random walk. We prove a large deviation principle for the area under the busy period of the Markov random walk, and we show that it exhibits a heavy-tailed behavior. Funding: The research of B. Zwart and M. Bazhba is supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek [Grant 639.033.413]. The research of J. Blanchet is supported by the National Science Foundation (NSF) [Grants 1915967, 1820942, and 1838576] as well as the Defense Advanced Research Projects Agency [Grant N660011824028]. The research of C.-H. Rhee is supported by the NSF [Grant CMMI-2146530].

On fluctuation-theoretic decompositions via Lindley-type recursions

Consider a Lévy process Y(t) over an exponentially distributed time Tβ with mean 1/β. We study the joint distribution of the running maximum Ȳ(Tβ) and the time epoch G(Tβ) at which this maximum last occurs. Our main result is a fluctuation-theoretic distributional equality: the vector (Ȳ(Tβ),G(Tβ)) can be written as a sum of two independent vectors, the first one being (Ȳ(Tβ+ω),G(Tβ+ω)) and the second one being the running maximum and corresponding time epoch under the restriction that the Lévy process is only observed at Poisson(ω) inspection epochs (until Tβ). We first provide an analytic proof for this remarkable decomposition, and then a more elementary proof that gives insight into the occurrence of the decomposition and into the fact that ω only appears in the right hand side of the decomposition. The proof technique underlying the more elementary derivation also leads to further generalizations of the decomposition, and to some fundamental insights into a generalization of the well known Lindley recursion.

On a modified version of the Lindley recursion

On a modified version of the Lindley recursion

A multiplicative version of the Lindley recursion

This paper presents an analysis of the stochastic recursion W_{i+1} = [V_iW_i+Y_i]^+ that can be interpreted as an autoregressive process of order 1, reflected at 0. We start our exposition by a discussion of the model’s stability condition. Writing Y_i=B_i-A_i, for independent sequences of nonnegative i.i.d. random variables {A_i}_{iin {mathbb N}_0} and {B_i}_{iin {mathbb N}_0}, and assuming {V_i}_{iin {mathbb N}_0} is an i.i.d. sequence as well (independent of {A_i}_{iin {mathbb N}_0} and {B_i}_{iin {mathbb N}_0}), we then consider three special cases (i) V_i equals a positive value a with certain probability pin (0,1) and is negative otherwise, and both A_i and B_i have a rational LST, (ii) V_i attains negative values only and B_i has a rational LST, (iii) V_i is uniformly distributed on [0, 1], and A_i is exponentially distributed. In all three cases, we derive transient and stationary results, where the transient results are in terms of the transform at a geometrically distributed epoch.

Improving routing decisions in parallel non-observable queues

We revisit the well-known problem of scheduling in $$N\ge 2$$ non-observable parallel single-server queues with one dispatcher having no queue to store the incoming jobs. The dispatcher does not observe the current states of the queues and servers and the sizes of the incoming jobs. The only available information to it is the job size distribution, job’s inter-arrival time distribution and server’s speeds. For this problem setting it is known that if the dispatcher can memorize the sequence of its previous decisions, then a deterministic policy is better than the random choice policy with respect to the job’s mean waiting and mean sojourn time. In this paper we address the following question: can the deterministic policy be improved if the dispatcher, in addition to the decisions made, can memorize also the times between the decisions? We describe the new policy and numerically show that it is almost always possible to do better with it than with the deterministic policy. Two algorithms for choosing decisions under the new policy are given. Both are based on the Lindley recursion and are approximate in nature, because utilize the discretization of the underlying distributions. Our findings show that the relative gain of the new policy may reach $$10\%$$ , when minimizing job’s mean sojourn time, and more than $$50\%$$ for the job’s mean waiting time.

Outpatient scheduling with unpunctual patients and no-shows

We assess appointment scheduling strategies for outpatient services. To this end, we consider a fixed-length consultation session in which K patients have to be scheduled at predefined appointment times, but who may not be punctual. Their effective arrival time deviates from their appointment time by a stochastic unpunctuality time. We assume general, possibly distinct distributions for the patients’ consultation times as well as for their unpunctuality. The heterogeneity of the consultation times is motivated by patient classification: the schedule can be adapted to the patients’ characteristics. Our evaluation approach is based on a modified Lindley recursion in a discrete-time framework and obtains accurate predictions for the moments of the patient waiting times as well as the doctor’s idle times and overtime. This evaluation method is then included in a local search algorithm to provide general insights into appointment scheduling under unpunctual patients. Our results suggest that the proposed method obtains substantial cost reductions when patient classification is correctly exploited. Finally, it is shown that our analysis can also be used to determine optimal sequencing rules for patients who arrive out of turn.

Estimating Loynes’ exponent

Loynes' distribution, which characterizes the one dimensional marginal of the stationary solution to Lindley's recursion, possesses an ultimately exponential tail for a large class of increment processes. If one can observe increments but does not know their probabilistic properties, what are the statistical limits of estimating the tail exponent of Loynes' distribution? We conjecture that in broad generality a consistent sequence of non-parametric estimators can be constructed that satisfies a large deviation principle. We present rigorous support for this conjecture under restrictive assumptions and simulation evidence indicating why we believe it to be true in greater generality.

Generalized Lindley-type recursive representations for multiserver tandem queues with blocking

Lindley's recursion is an explicit recursive equation that describes the recursive relationship between consecutive waiting times in a single-stage single-server queue. In this paper, we develop explicit recursive representations for multiserver tandem queues with blocking. We demonstrate the application of these recursive representations with simulations of large-scale tandem queueing networks. We compare the computational efficiency of these representations with two other popular discrete-event simulation approaches, namely, event scheduling and process interaction. Experimental results show that these representations are seven (or more) times faster than their counterparts based on the event-scheduling and process-interaction approaches.

Tail Asymptotics for a Random Sign Lindley Recursion

We investigate the tail behaviour of the steady-state distribution of a stochastic recursion that generalises Lindley's recursion. This recursion arises in queueing systems with dependent interarrival and service times, and includes alternating service systems and carousel storage systems as special cases. We obtain precise tail asymptotics in three qualitatively different cases, and compare these with existing results for Lindley's recursion and for alternating service systems.

Tail Asymptotics for a Random Sign Lindley Recursion

We investigate the tail behaviour of the steady-state distribution of a stochastic recursion that generalises Lindley's recursion. This recursion arises in queueing systems with dependent interarrival and service times, and includes alternating service systems and carousel storage systems as special cases. We obtain precise tail asymptotics in three qualitatively different cases, and compare these with existing results for Lindley's recursion and for alternating service systems.

ON THE LAW OF THE iTH WAITING TIME IN A BUSY PERIOD OF G/M/c QUEUES

We use induction to derive the distribution of the waiting time of the ith waiting customer in a busy period for a G/M/1 queue with a first come–first serve service. A trivial implication gives the law for the ith waiting time in a busy period for a G/M/c queue. Finally, we use the Lindley recursion to relate our results to the distribution of random walks.

Time-Dependent Behaviour of an Alternating Service Queue

We consider a model describing the waiting time of a server alternating between two service points. This model is described by a Lindley-type equation. We are interested in the time-dependent behaviour of this system and derive explicit expressions for its time-dependent waiting-time distribution, the correlation between waiting times, and the distribution of the cycle length. As our model is closely related to Lindley's recursion, we compare our results to those derived for Lindley's recursion.

Large deviations of the sojourn time for queues in series

We consider an open queueing network consisting of an arbitrary number of queues in series. We assume that the arrival process into the first queue and the service processes at the individual queues are jointly stationary and ergodic, and that the mean inter-arrival time exceeds the mean service time at each of the queues. Starting from Lindley's recursion for the waiting time, we obtain a simple expression for the total delay (sojourn time) in the system. Under some mild additional assumptions, which are satisfied by many commonly used models, we show that the delay distribution has an exponentially decaying tail and compute the exact decay rate. We also find the most likely path leading to the build-up of large delays. Such a result is of relevance to communication networks, where it is often necessary to guarantee bounds on the probability of large delays. Such bounds are part of the specification of the quality of service desired by the network user.

A new approach to the Lindley recursion

We consider the standard Lindley recursion for integer-valued random variables. A new method for determining the corresponding distributions is presented which in the case when the involved random variables are bounded from below, say by − K, K ϵ K, reduces to the solution of a ( K × K)-system of linear equations. The stationary distribution is also obtained by this approach.

On Whitt's conjecture for queues in which service times and interarrival times depend linearly and randomly upon waiting times

We consider a modification of the standardG/G/1 queueing system with infinite waiting space and the first-in-first-out discipline in which the service times and interarrival times depend linearly and randomly on the waiting times. In this model the waiting times satisfy a modified version of the classical Lindley recursion. When the waiting-time distributions converge to a proper limit, Whitt [10] proposed a normal approximation for this steady-state limit. In this paper we prove a limit theorem for the steady-state limit of the system. Thus, our result provides a solid foundation for Whitt's normal approximation of the steady-state distribution of the system.

Queues with service times and interarrival times depending linearly and randomly upon waiting times

We consider a modification of the standardG/G/1 queue with unlimited waiting space and the first-in first-out discipline in which the service times and interarrival times depend linearly and randomly on the waiting times. In this model the waiting times satisfy a modified version of the classical Lindley recursion. We determine when the waiting-time distributions converge to a proper limit and we develop approximations for this steady-state limit, primarily by applying previous results of Vervaat [21] and Brandt [4] for the unrestricted recursionYn+1=CnYn+Xn. Particularly appealing for applications is a normal approximation for the stationary waiting time distribution in the case when the queue only rarely becomes empty. We also consider the problem of scheduling successive interarrival times at arrival epochs, with the objective of achieving nearly maximal throughput with nearly bounded waiting times, while making the interarrival time sequence relatively smooth. We identify policies depending linearly and deterministically upon the work in the system which meet these objectives reasonably well; with these policies the waiting times are approximately contained in a specified interval a specified fraction of time.