Independent Renewal Processes Research Articles

On Some Stochastic Ordering Comparisons for Renewal Processes

After a brief introduction to ordinary renewal process, we have defined compound renewal process generated by an ordinary renewal process followed by the distributional properties of the corresponding random variables. A few results are obtained by comparing independent renewal processes with respect to several stochastic orderings between the generating inter-arrival time random variables, like, stochastic order, hazard rate order, likelihood ratio order and variability order, as well as some ageing classes of the generating random variables. Some numerical illustrations are given. The results obtained here appear to be new.

A New Upper Bound on Cache Hit Probability for Non-Anticipative Caching Policies

Caching systems have long been crucial for improving the performance of a wide variety of network and web-based online applications. In such systems, end-to-end application performance heavily depends on the fraction of objects transferred from the cache, also known as the cache hit probability . Many caching policies have been proposed and implemented to improve the hit probability. In this work, we propose a new method to compute an upper bound on hit probability for all non-anticipative caching policies and for policies that have no knowledge of future requests. Our key insight is to order the objects according to the ratio of their Hazard Rate (HR) function values to their sizes, and place in the cache the objects with the largest ratios till the cache capacity is exhausted. When object request processes are conditionally independent, we prove that this cache allocation based on the HR-to-size ratio rule guarantees the maximum achievable expected number of object hits across all non-anticipative caching policies. Further, the HR ordering rule serves as an upper bound on cache hit probability when object request processes follow either independent delayed renewal process or a Markov modulated Poisson process. We also derive closed form expressions for the upper bound under some specific object request arrival processes. We provide simulation results to validate its correctness and to compare it to the state-of-the-art upper bounds, such as produced by Bélády’s algorithm. We find it to be tighter than state-of-the-art upper bounds for some specific object request arrival processes such as independent renewal, Markov modulated, and shot noise processes.

The Age of Information in Networks: Moments, Distributions, and Sampling

A source provides status updates to monitors through a network with state defined by a continuous-time finite Markov chain. An age of information (AoI) metric is used to characterize timeliness by the vector of ages tracked by the monitors. Based on a stochastic hybrid systems (SHS) approach, first order linear differential equations are derived for the temporal evolution of both the moments and the moment generating function (MGF) of the age vector components. It is shown that the existence of a non-negative fixed point for the first moment is sufficient to guarantee convergence of all higher order moments as well as a region of convergence for the stationary MGF vector of the age. The stationary MGF vector is then found for the age on a line network of preemptive memoryless servers. From this MGF, it is found that the age at a node is identical in distribution to the sum of independent exponential service times. This observation is then generalized to linear status sampling networks in which each node receives samples of the update process at each preceding node according to a renewal point process. For each node in the line, the age is shown to be identical in distribution to a sum of independent renewal process age random variables.

Quantitative and qualitative analysis of asynchronous neural activity

The activity of a sparse network of leaky integrate-and-fire neurons is carefully revisited with reference to a regime of a bona-fide asynchronous dynamics. The study is preceded by a finite-size scaling analysis, carried out to identify a setup where collective synchronization is negligible. The comparison between quenched and annealed networks reveals the emergence of substantial differences when the coupling strength is increased, via a scenario somehow reminiscent of a phase transition. For sufficiently strong synaptic coupling, quenched networks exhibit a highly bursting neural activity, well reproduced by a self-consistent approach, based on the assumption that the input synaptic current is the superposition of independent renewal processes. The distribution of interspike intervals turns out to be relatively long-tailed; a crucial feature required for the self-sustainment of the bursting activity in a regime where neurons operate on average (much) below threshold. A semi-quantitative analogy with Ornstein-Uhlenbeck processes helps validating this interpretation. Finally, an alternative explanation in terms of Poisson processes is offered under the additional assumption of mutual correlations among excitatory and inhibitory spikes.

Modeling LRU cache with invalidation

Least Recently Used (LRU) is a very popular caching replacement policy. It is very easy to implement and offers good performance, especially when data requests are temporally correlated, as in the case of web traffic.When the data content can change during time, as in the case of dynamic websites or within databases, there is the need to prevent the cache to serve stale data. This is usually done by triggering an invalidation event in the cache, to purge all the previously cached data concerning the invalidated data item. The invalidation process tends to worsen the caching performance, since stored items can be invalidated after a short time, thus wasting storage space.Several models in the literature allow quantifying the cache hit probability of an LRU cache, but, to the best of our knowledge, the presence of invalidation events has not been taken into account so far.In this paper, we present an analytical performance evaluation of LRU caches that takes into account data requests and invalidation events, both modeled as independent renewal processes. Simulation results show the accuracy of our model. Moreover, we apply our model to evaluate the LRU performance in the case of a real application, Wikipedia. Finally, we evaluate by means of simulations the effect of invalidation in hierarchical caching.Our work allows us to conclude that the presence of invalidation events does not severely impact the LRU performance in single caches. As a matter of fact, invalidation effects can be ignored there, unless the invalidation rate is comparable with the request rate and the per-object invalidation rate and request rate are highly correlated. However, in the case of hierarchical caching, even a limited effect of invalidation on first-level caches is sufficient to noticeably affect the performance of second level/downstream caches.

Backtracking and Mixing Rate of Diffusion on Uncorrelated Temporal Networks

We consider the problem of diffusion on temporal networks, where the dynamics of each edge is modelled by an independent renewal process. Despite the apparent simplicity of the model, the trajectories of a random walker exhibit non-trivial properties. Here, we quantify the walker’s tendency to backtrack at each step (return where he/she comes from), as well as the resulting effect on the mixing rate of the process. As we show through empirical data, non-Poisson dynamics may significantly slow down diffusion due to backtracking, by a mechanism intrinsically different from the standard bus paradox and related temporal mechanisms. We conclude by discussing the implications of our work for the interpretation of results generated by null models of temporal networks.

Heavy traffic analysis of roving server networks

ABSTRACTThis article studies the heavy-traffic (HT) behavior of queueing networks with a single roving server. External customers arrive at the queues according to independent renewal processes and after completing service, a customer either leaves the system or is routed to another queue. This type of customer routing in queueing networks arises very naturally in many application areas (in production systems, computer- and communication networks, maintenance, etc.). In these networks, the single most important characteristic of the system performance is oftentimes the path time, i.e., the total time spent in the system by an arbitrary customer traversing a specific path. The current article presents the first HT asymptotic for the path-time distribution in queueing networks with a roving server under general renewal arrivals. In particular, we provide a strong conjecture for the system’s behavior under HT extending the conjecture of Coffman et al.[8,9] to the roving server setting of the current article. By combining this result with novel light-traffic asymptotics, we derive an approximation of the mean path time for arbitrary values of the load and renewal arrivals. This approximation is not only highly accurate for a wide range of parameter settings, but is also exact in various limiting cases.

Local asymptotics for the first intersection of two independent renewals

We study the intersection of two independent renewal processes, ρ = τ ∩σ. Assuming that P(τ 1 = n) = ϕ(n) n −(1+α) and P(σ 1 = n) = ϕ(n) n −(1+ α) for some α, α 0 and some slowly varying ϕ, ϕ, we give the asymptotic behavior first of P(ρ 1 > n) (which is straightforward except in the case of min(α, α) = 1) and then of P(ρ 1 = n). The result may be viewed as a kind of reverse renewal theorem, as we determine probabilities P(ρ 1 = n) while knowing asymptotically the renewal mass function P(n ∈ ρ) = P(n ∈ τ)P(n ∈ σ). Our results can be used to bound coupling-related quantities, specifically the increments |P(n ∈ τ) − P(n − 1 ∈ τ)| of the renewal mass function. 1. Intersection of two independent renewals We consider two independent (discrete) renewal processes τ and σ, whose law are denoted respectively P τ and P σ , and the renewal process of intersections, ρ = τ ∩ σ. We denote P = P τ ⊗ P σ. The process ρ appears in various contexts. In pinning models, for example, it may appear directly in the definition of the model (as in [1], where σ represents sites with nonzero disorder values, and τ corresponds to the polymer being pinned) or it appears in the computation of the variance of the partition function via a replica method (see for example [20]), and is central in deciding whether disorder is relevant or irrelevant in these models, cf. [3]. When τ and σ have the same inter-arrival distribution, ρ 1 is related to the coupling time of τ and σ, if we allow τ and σ to start at different points. In particular, in the case µ := E[τ 1 ] < +∞, the coupling time ρ 1 has been used to study the rate of convergence in the renewal theorem, see [16, 17], using that

Diffusion Models for Double-Ended Queues with Renewal Arrival Processes

We study a double-ended queue where buyers and sellers arrive to conduct trades. When there is a pair of buyer and seller in the system, they immediately transact a trade and leave. Thus there cannot be a non-zero number of buyers and sellers simultaneously in the system. We assume that sellers and buyers arrive at the system according to independent renewal processes, and they would leave the system after independent exponential patience times. We establish fluid and diffusion approximations for the queue length process under a suitable asymptotic regime. The fluid limit is the solution of an ordinary differential equation, and the diffusion limit is a time-inhomogeneous asymmetric Ornstein-Uhlenbeck process (O-U process). A heavy traffic analysis is also developed, and the diffusion limit in the stronger heavy traffic regime is a time-homogeneous asymmetric O-U process. The limiting distributions of both diffusion limits are obtained. We also show the interchange of the heavy traffic and steady state limits.

A clearing system with impatient passengers: asymptotics and estimation in a bus stop model

At some location buses stop to pick up the passengers waiting there and leave immediately, empty or occupied. Passenger arrival times as well as bus arrival times form independent renewal processes. Every passenger is willing to wait for a random amount of time before leaving, and every bus takes away all waiting passengers. For this pickup model, we study the distributions of the number of waiting passengers and of the individual sojourn times. The sojourn times lead to a Markov chain embedded in the superposition of the two underlying renewal arrival processes, for which we study its convergence toward stationarity.

Stability of networked control systems with asynchronous renewal links: An impulsive systems approach

We consider networked control systems in which sensors, actuators, and controller transmit through asynchronous communication links, each introducing independent and identically distributed intervals between transmissions. We model these scenarios through impulsive systems with several reset maps triggered by independent renewal processes, i.e., the intervals between jumps associated with a given reset map are identically distributed and independent of the other jump intervals. For linear dynamic and reset maps, we establish that mean exponential stability is equivalent to the spectral radius of an integral operator being less than one. We also prove that the origin of a non-linear impulsive system is (locally) stable with probability one if its local linearization about the zero equilibrium is mean exponentially stable, which justifies the importance of studying the linear case. The applicability of the results is illustrated through an example using a linearized model of a batch-reactor.

Optimal dividend-payout in random discrete time

Abstract Assume that the surplus process of an insurance company is described by a general Lévy process and that possible dividend pay-outs to shareholders are restricted to random discrete times which are determined by an independent renewal process. Under this setting we show that the optimal dividend pay-out policy is a band-policy. If the renewal process is a Poisson process, it is further shown that for Cramér–Lundberg risk processes with exponential claim sizes and its diffusion limit the optimal policy collapses to a barrier-policy. Finally, a numerical example is given for which the optimal bands can be calculated explicitly. The random observation procedure studied in this paper also allows for an interpretation in terms of a random walk model with a certain type of random discounting.

Modeling Cross Correlation in Three-Moment Four-Parameter Decomposition Approximation of Queueing Networks

In two-moment decomposition approximations of queueing networks, the arrival process is modeled as a renewal process, and each station is approximated as a GI/G/1 queue whose mean waiting time is approximated based on the first two moments of the interarrival times and the service times. The departure process is also approximated as a renewal process even though the autocorrelation of this process may significantly affect the performance of the subsequent queue depending on the traffic intensity. When the departure process is split into substreams by Markovian random routing, the split processes typically are modeled as independent renewal processes even though they are correlated with each other. This cross correlation might also have a serious impact on the queueing performance. In this paper, we propose an approach for modeling both the cross correlation and the autocorrelation by a three-moment four-parameter decomposition approximation of queueing networks. The arrival process is modeled as a nonrenewal process by a two-state Markov-modulated Poisson process, viz., MMPP(2). The cross correlation between randomly split streams is accounted for in the second and third moments of the merged process by the innovations method. The main contribution of the present research is that both the cross correlation and the autocorrelation can be modeled in parametric decomposition approximations of queueing networks by integrating the MMPP(2) approximation of the arrival/departure process and the innovations method. We also present numerical results that strongly support our refinements.

Bayesian Analysis of a Superimposed Renewal Process

Stochastic behavior of an aero-engine failure/repair process has been analyzed from a Bayesian perspective. Number of failures/repairs in the component-sockets of this multi-component system are assumed to follow independent renewal processes with Weibull inter-arrival times. Based on the field failure/repair data of a large number of such engines and independent Gamma priors on the scale parameters and log-concave priors on the shape parameters, an exact method of sampling from the resulting posterior distributions of the parameters has been proposed. These generated parameter values are next utilised in obtaining the posteriors of the expected number of system repairs, system failure rate, and the conditional intensity function, which are computed using a recursive formula.

Quenched large deviation principle for words in a letter sequence

When we cut an i.i.d. sequence of letters into words according to an independent renewal process, we obtain an i.i.d. sequence of words. In the annealed large deviation principle (LDP) for the empirical process of words, the rate function is the specific relative entropy of the observed law of words w.r.t. the reference law of words. In the present paper we consider the quenched LDP, i.e., we condition on a typical letter sequence. We focus on the case where the renewal process has an algebraic tail. The rate function turns out to be a sum of two terms, one being the annealed rate function, the other being proportional to the specific relative entropy of the observed law of letters w.r.t. the reference law of letters, with the former being obtained by concatenating the words and randomising the location of the origin. The proportionality constant equals the tail exponent of the renewal process. Earlier work by Birkner considered the case where the renewal process has an exponential tail, in which case the rate function turns out to be the first term on the set where the second term vanishes and to be infinite elsewhere. In a companion paper the annealed and the quenched LDP are applied to the collision local time of transient random walks, and the existence of an intermediate phase for a class of interacting stochastic systems is established.

Characterizing the miss sequence of the LRU cache

Renewed interest in caching systems stems from their wide-spread use for reducing the document download latency over the Internet. Since caches are usually organized in a hierarchical manner, it is important to study the performance properties of tandem caches. The first step in understanding this problem is to characterize the miss stream from one single cache since it represents the input to the next level cache. In this regard, we discover that the miss stream from one single cache is approximated well by the superposition of a number of asymptotically independent renewal processes. Interestingly, when this weakly correlated miss sequence is fed into another cache, this barely observable correlation can lead to measurably different caching performance when compared to the independent reference model. This result is likely to enable the development of a rigorous analysis of the tandem cache performance.

A note on random coin tossing

Harris and Keane [Probab. Theory Related Fields 109 (1997) 27-37] studied absolute continuity/singularity of two probabilities on the coin-tossing space, one representing independent tosses of a fair coin, while in the other a biased coin is tossed at renewal times of an independent renewal process and a fair coin is tossed at all other times. We extend their results by allowing possibly different biases at the different renewal times. We also investigate the contiguity and asymptotic separation properties in this kind of set-up and obtain some sufficient conditions. Keywords: renewal process, absolute continuity, singularity, contiguity, asymptotic separation, martingale convergence theorem

Conditional large deviations for a sequence of words

Cut an i.i.d. sequence ( X i ) of ‘letters’ into ‘words’ according to an independent renewal process. Then one obtains an i.i.d. sequence of words, and thus the level 3 large deviation behaviour of this sequence of words is governed by the specific relative entropy. We consider the corresponding problem for the conditional empirical process of words, where one conditions on a typical underlying ( X i ) . We find that if the tails of the word lengths decay exponentially, the large deviations under the conditional distribution are almost surely again governed by the specific relative entropy, but the set of attainable limits is restricted. We indicate potential applications of such a conditional LDP to the computation of the quenched free energy for directed polymer models with random disorder.

Superposition of renewal processes with heavy-tailed interarrival times

Limiting distributions for the spent and residual life times in a superposition of independent renewal processes are obtained in the case when the interarrival times belong to domains of attraction of stable laws.

Runs in superpositions of renewal processes with applications to discrimination

Wald and Wolfowitz [Ann. Math. Statist. 11 (1940) 147–162] introduced the run test for testing whether two samples of i.i.d. random variables follow the same distribution. Here a run means a consecutive subsequence of maximal length from only one of the two samples. In this paper we contribute to the problem of runs and resulting test procedures for the superposition of independent renewal processes which may be interpreted as arrival processes of customers from two different input channels at the same service station. To be more precise, let ( S n ) n ⩾ 1 and ( T n ) n ⩾ 1 be the arrival processes for channel 1 and channel 2, respectively, and ( W n ) n ⩾ 1 their be superposition with counting process N ( t ) = def sup { n ⩾ 1 : W n ⩽ t } . Let further R n * be the number of runs in W 1 , … , W n and R t = R N ( t ) * the number of runs observed up to time t. We study the asymptotic behavior of R n * and R t , first for the case where ( S n ) n ⩾ 1 and ( T n ) n ⩾ 1 have exponentially distributed increments with parameters λ 1 and λ 2 , and then for the more difficult situation when these increments have an absolutely continuous distribution. These results are used to design asymptotic level α tests for testing λ 1 = λ 2 against λ 1 ≠ λ 2 in the first case, and for testing for equal scale parameters in the second.