Stochastics and Statistics
The intercept term of the asymptotic variance curve for some queueing output processes

https://doi.org/10.1016/j.ejor.2014.10.051Get rights and content

Highlights

  • Explicit analysis of variance curves of queueing output processes.

  • Explicit analysis of the covariance between queue lengths and flows.

  • Role of the deviation matrix in formulas related to Markovian Point Processes.

Abstract

We consider the output processes of some elementary queueing models such as the M/M/1/K queue and the M/G/1 queue. An important performance measure for these counting processes is their variance curve v(t), which gives the variance of the number of customers in the time interval [0, t]. Recent work has revealed some non-trivial properties dealing with the asymptotic rate at which the variance curve grows. In this paper we add to these results by finding explicit expressions for the intercept term of the linear asymptote. For M/M/1/K queues our results are based on the deviation matrix of the generator. It turns out that by viewing output processes as Markovian Point Processes and considering the deviation matrix, one can obtain explicit expressions for the intercept term, together with some further insight regarding the BRAVO (Balancing Reduces Asymptotic Variance of Outputs) effect. For M/G/1 queues our results are based on a classic transform of D. J. Daley. In this case we represent the intercept term of the variance curve in terms of the first three moments of the service time distribution. In addition we shed light on a conjecture of Daley, dealing with characterization of stationary M/M/1 queues within the class of stationary M/G/1 queues, based on the variance curve.

Introduction

Many models in applied probability and stochastic operations research involve counting processes. Such processes occur in supply chains, health care systems, communication networks as well as many other contexts involving service, logistics and/or technology. The canonical counting process example is the Poisson process. Generalizations include renewal processes, Markovian Point Processes (see for example Latouche and Ramaswami, 1999, Section 3.5 or Asmussen, 2003, Section XI.1), or general simple point processes on the line (see for example Daley and Vere-Jones, 2003).

Sometimes counting processes are used in their own right, while at other times they constitute components of more complicated models such as queues, population processes or risk models. In other instances, counting processes are implicitly defined and constructed through applied probability models. For example, a realization of a queue induces additional counting processes such as the departure process, {D(t), t ≥ 0}, counting the number of serviced customers in the queue until time t.

Departure counting processes of queues have been heavily studied in applied probability and operations research. Classic applied probability surveys are Daley (1976) and Disney and Konig (1985). More recent studies in operations research are Hendricks (1992),Tan (1999) and Tan (1997) where the authors consider departures in and within manufacturing production lines. Indeed, from an operational viewpoint, quantification of the variability of flows within a network is key. A similar comment applies to the flows of finished products at the end of the production process. From a theoretical perspective, there remain some open questions about the ability to characterize {D(t)} as a Markovian Point Process, as in Bean and Green (2000),Bean et al. (1998) and Olivier and Walrand (1994). Further, the discovery of the BRAVO effect (Balancing Reduces Asymptotic Variance of Outputs) has motivated research on the variability of departure processes of queues, particularly in critically loaded regimes. Recent papers on this topic are Al Hanbali et al. (2011),Daley (2011),Daley et al. (2014),Nazarathy (2011) and Nazarathy and Weiss (2008).

Next to the mean curve, m(t)=E[D(t)], an almost equally important performance measure of a counting processes is the variance curve, v(t) = Var(D(t)). For example, for a Poisson process with rate α, the variance curve v(t)=αtis the same as the mean curve. For more complicated counting processes, the variance curve is not as simple and is not the same as the mean curve. For example, for a stationary (also known as equilibrium) renewal-process with inter-renewal times distributed as the sum of two independent exponential random variables, each with mean (2α)−1, we have m(t)=αt14+14e4αt,v(t)=α12t+1818e4αt.For the ordinary case of the same renewal process (the first inter-renewal time is distributed as all the rest) the variance curve is v(t)=α12t+116te4αt116e8αt.These explicit examples are taken from Cox (1962, Section 4.5). In fact, for general, non-lattice, renewal processes (both equilibrium and ordinary), with inter-renewal times having a finite second moment, with squared coefficient of variation c2, and mean α−1, it is well known that, v(t)=αc2t+o(t),as t → ∞ (which is the limiting regime used throughout this paper). However, in general, a finer description of v(t) (through the o(t) term) is typically not as simple as in the examples above.

If the third moment of the inter-renewal time is finite, then v(t)={αc2t+54(c41)23(γc32)+o(1),fortheequilibriumcase,αc2t+12(c41)13(γc32)+o(1),fortheordinarycase,where γ is the skewness coefficient of the inter-renewal time.1 We remind the reader that for exponential random variables (making the renewal process a Poisson process), c2 = 1 and γ = 2, and the ordinary and equilibrium versions of a Poisson process are identical. See Asmussen (2003) and Daley and Vere-Jones (2003) for more background on renewal processes. Eq. (2) appears under a slightly different representation in Cox (1962) and was essentially first found in Smith (1959). Generalizations of renewal processes are in Brown and Solomon (1975),Daley and Mohan (1978) and Hunter (1969).

The above examples indicate that, for counting processes in general, it is likely to be fruitful to look for an asymptotic expression for the variance curve of the form v(t)=v¯t+b¯+o(1).We refer to v¯ as the asymptotic variance rate and to b¯ as the intercept term. A point to observe is that, for a renewal process, b¯ depends on the version of the renewal process (ordinary vs. equilibrium) while v¯ does not. Since the latter depends on the initial conditions, we generally employ the notation b¯e for the stationary (equilibrium) system, b¯0 for systems starting empty and b¯θ for systems with arbitrary initial conditions.

Moving on from renewal processes to implicitly defined counting processes, the variance curve is typically more complicated to describe and characterize. For example, while the output of a stationary M/M/1 queue with arrival rate λ and service rate μ is simply a Poisson process with rate λ (see Kelly, 1979), the variance curve when the system starts empty at time 0 is much more complicated than v(t) = λt. It can be represented in terms of integrals of expressions involving Bessel functions of the first kind, and requires several lines to be written out fully (as in Theorem 5.1 of Al Hanbali et al., 2011). Nevertheless (see Theorem 5.2 in Al Hanbali et al., 2011) the curve can be sensibly approximated as follows: v(t)={λtρ(1ρ)2+o(1),ifρ<1,2(12π)λtλπt1/2+π24π+o(1),ifρ=1,μtρ(1ρ)2+o(1),ifρ>1,where ρ = λ/μ.

As observed from the formula above, it may be initially quite surprising that the asymptotic variance rate is reduced by a factor of 2(1 − 2/π) ≈ 0.73 when ρ changes from being approximately 1 to exactly 1. This is a manifestation of the BRAVO effect. BRAVO was first observed for M/M/1/K queues in Nazarathy and Weiss (2008) in which case, as K → ∞, the factor is 2/3, a fact that we confirm in this paper. It was later analyzed for M/M/1 queues and more generally GI/G/1 queues in Al Hanbali et al. (2011). BRAVO has been numerically conjectured for GI/G/1/K queues in Nazarathy (2011), and observed for multi-server M/M/s/K queues in the many-server scaling regime in Daley et al. (2014).

Our focus in this paper is on the more subtle intercept term b¯. For a stationary M/M/1 queue, {D(t)} is a Poisson process and thus b¯e=0. As opposed to that, for an M/M/1 queue starting empty, it follows from (4) that b¯0=ρ/(1ρ)2 as long as ρ ≠ 1. When ρ = 1, we see from (4) that the variance curve does not have the asymptotic form (3). This can happen more generally. If, for example, there is sufficient long range dependence in the counting process, then the variance can grow super-linearly (see Daley and Vesilo, 1997 for some examples). This demonstrates that the asymptotic variance rate, v¯, and the intercept term, b¯, need not exist for every counting process. Nevertheless, for a variety of models and situations, both v¯ and b¯ exist, and thus the linear asymptote is well-defined. In such cases, having a closed formula is beneficial for performance analysis of the model at hand.

We are now faced with the challenge of finding the intercept term for other counting processes generated by queues. In this paper we carry out such an analysis for two models related to the M/M/1 queue: a finite capacity M/M/1/K queue, and an infinite capacity M/G/1 queue. Besides obtaining explicit formulas for b¯e,b¯0 and b¯θ, our investigation also pinpoints some of the analytical challenges involved and raises some open questions. Here is a summary of our main contributions.

In this case the departure process is a Markovian Point Process. The linear asymptote is then given by formulas based on the matrix Λ = (1πΛ)−1, where Λ is the generator matrix of the (finite) birth-death process, π is its stationary distribution taken as a row vector, and 1 is a column vector of 1’s. In the case where ρ = 1, the distribution π is uniform and an explicit expression for Λ was previously found, which in turn yielded the equilibrium version of the intercept term, b¯e=7K4+28K3+37K2+18K180(K+1)2,in Proposition 4.4 of Nazarathy and Weiss (2008). When ρ ≠ 1, the form of the inverse Λ is more complicated and an expression for b¯ has not been previously known. We are now able to find such an expression for both the stationary version and for arbitrary initial conditions. Our results are based on relating Λ to the matrix Λ=0(P(t)1π)dt,where P( · ) is the transition probability kernel of the birth-death process. The matrix Λ is called the deviation matrix (also known as the Drazin inverse of − Λ), and we are able to provide explicit expressions for the entries of this matrix. Our contribution also includes some useful results regarding the asymptotic covariance between the count and phase in arbitrary Markovian Point Processes which, to the best of our knowledge, have not appeared elsewhere.

When the third moment of the service time distribution is finite, then it is known that the stationary queue length has a finite variance. When the service time distribution is not exponential, the form of b¯ has not previously been known. Our contribution is an exact expression for the b¯ term based on the first three moments of G. We begin with b¯e, after which we employ a simple coupling argument to find b¯θ and b¯0.

The structure of the rest of the paper is as follows: In Section 2 we present our M/M/1/K queue results for b¯ together with a discussion of the deviation matrix and its application to Markovian Point Processes. In Section 3 we present our M/G/1 queue results for b¯, and discuss a related conjecture of Daley, dealing with a characterization of the M/M/1 queue within the class of stationary M/G/1 queues. We conclude in Section 4.

Section snippets

The M/M/1/K queue

We begin our investigation with the M/M/1/K queue, where K denotes the total capacity of the system. In this case, it is well known that the departure process {D(t)} is a Markovian Point Process and is a renewal processes only when K = 1 or K = ∞. Some standard references on Markovian Point Processes are Asmussen (2003, Section XI.1) and Latouche and Ramaswami (1999, Section 3.5).

Denote the arrival rate by λ > 0, the service rate by μ > 0 and let ρ = λ/μ be the traffic intensity. The queue

The M/G/1 queue

We now consider the departure process of the M/G/1 queue. In this case, the departure process {D(t)} is generally not a Markovian Point Process, and the analysis is more complicated. Nevertheless we are able to obtain some partial results about the linear asymptote of v(t). Our approach is to first assume the existence of a linear asymptote and then to find an elegant formula for the intercept term under this assumption, generalizing the expression for the M/M/1 queue in (4) for the stable

Conclusion

In going through the detailed Markovian Point Process derivations for M/M/1/K queues, we have illustrated how asymptotic quantities such as b¯ may be obtained explicitly. The key is to have explicit expressions for mean hitting times in the underlying Markov chain. By using the deviation matrix we have gained some further insight into the BRAVO effect. In plotting the graphs of d¯v and d¯b, we observe that spikes occur when λμ in a similar fashion to the BRAVO effect.

For the M/G/1 queue, the

Acknowledgments

We thank two anonymous referees for their comments. We also thank Onno Boxma, Brian Fralix and Guy Latouche for useful discussions and advice. Yoni Nazarathy is supported by Australian Research Council (ARC) grants DP130100156 and DE130100291. Yoav Kerner is supported by Israeli Science Foundation (ISF) grant 1319/11. Yoav Kerner and Yoni Nazarathy also thank EURANDOM for hosting and support. Sophie Hautphenne and Peter Taylor are supported by Australian Research Council (ARC) grant DP110101663

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