# Robust heavy-traffic approximations for service systems facing overdispersed demand

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## Abstract

Arrival processes to service systems often display fluctuations that are larger than anticipated under the Poisson assumption, a phenomenon that is referred to as *overdispersion*. Motivated by this, we analyze a class of discrete-time stochastic models for which we derive heavy-traffic approximations that are scalable in the system size. Subsequently, we show how this leads to novel capacity sizing rules that acknowledge the presence of overdispersion. This, in turn, leads to robust approximations for performance characteristics of systems that are of moderate size and/or may not operate in heavy traffic.

## Keywords

Heavy-traffic approximations Overdispersion Saddle point method Random walk## Mathematics Subject Classification

60K25 60G50 30E20 41A60## 1 Introduction

One of the most prevalent assumptions in queueing theory is the assumption that the number of arrivals over any given period is a Poisson random variable with deterministic rate, whose variance equals its expectation. Although natural and convenient from a mathematical viewpoint, the Poisson assumption often fails to be confirmed in practice. Namely, a growing number of empirical studies show that the variance of demand typically deviates from the mean significantly. Recent work [24, 26] reports variance being strictly less than the mean in health care settings employing appointment booking systems. This reduction of variability can be accredited to the goal of the booking system to create a more predictable arrival pattern. On the other hand, in other scenarios with no control over the arrivals, the variance can dominate the mean; see [4, 5, 6, 11, 12, 17, 19, 23, 25, 30, 31, 34, 38, 41]. The feature that variability is higher than one expects from the Poisson assumption is referred to as *overdispersion* and serves as the primary motivation for this work.

Stochastic models with the Poisson assumption have been widely applied to optimize capacity levels in service systems. When stochastic models, however, do not take into account overdispersion, resulting performance estimates are likely to be overoptimistic. The system then ends up being underprovisioned, which possibly causes severe performance problems, particularly in critical loading.

A significant part of the queueing literature has focused on extending Poisson arrival processes to more bursty arrival processes, and analyzing these models using, for example, matrix-analytic models [29, 33]. In this paper, we focus on a different cause of overdispersion in arrival processes, which is *arrival rate uncertainty*. Since model primitives, in particular the arrival rate, are typically estimated through historical data, these are prone to be subject to forecasting errors. In the realm of Poisson processes, this inherent uncertainty can be acknowledged by viewing the arrival rate \(\Lambda _n\) itself as being stochastic. The resulting doubly stochastic Poisson process, also known as a Cox process (first presented in [14]), implies that demand in a given interval \(A_{k,n}\) follows a mixed Poisson distribution. In this case, the expected demand per period equals \(\mu _n = {\mathbb {E}}[\Lambda _n]\), while the variance is \(\sigma _n^2 = {\mathbb {E}}[\Lambda _n]+\mathrm{Var}\,\Lambda _n\). By selecting the distribution of the mixing factor \(\Lambda _n\), the magnitude of overdispersion can be made arbitrarily large, and only a deterministic \(\Lambda _n\) leads to a true Poisson process.

The mixed Poisson model presents a useful way to fit both the mean and variance to real data, particularly in case of overdispersion. The mixing distribution can be estimated parametrically or nonparametrically; see [23, 30]. A popular parametric family is the Gamma distribution, which gives rise to an effective data fitting procedure that uses the fact that a Gamma mixed Poisson random variable follows a negative binomial distribution. We will in this paper adopt the assumption of a Gamma–Poisson mixture as the demand process.

We investigate the impact of this modeling assumption within the context of a classical model in queueing theory, which is the reflected random walk. In particular, we consider a sequence of such random walks, indexed by *n*, with increments \(A_{k,n}-s_n\), where \(A_{k,n}\sim \,\mathrm{Pois}(\Lambda _n)\) and \(s_n\) denotes the system capacity, and we consider a regime in which the system approaches heavy traffic. We are especially interested in the impact of overdispersion on the way performance measures scale, and how they impact capacity allocation rules.

The first part of our analysis relates to [37], in which a sequence of cyclically thinned queues, denoted by \(G_n/G_n/1\) queues, is considered. Here, \(G_n\) indicates that only every \(n\mathrm{th}\) point of the original point process is considered. In this framework, it is shown that the stationary waiting time can be characterized as the maximum of a random walk, in which the increments grow indefinitely. Under appropriate heavy-traffic scaling, the authors prove convergence to a Gaussian random walk and moreover characterize the limits of the stationary waiting time moments. Our work differs with respect to [37] in the sense that we study a discrete-time model, rather than the continuous-time \(G_n/G_n/1\) queue. Also, the presence of the overdispersion requires us to employ an alternative scaling.

Furthermore, our approach through Pollaczek’s formula allows us to derive estimates for performance measures in pre-limit, i.e., large but finite-size, systems. Mathematically, this second part of our analysis is related to previous work [22]. In particular, we use a refinement of the saddle point technique to establish our asymptotic estimates. The associated analysis is substantially more involved in the present situation, as we will explain in Sect. 4.

*Structure of the paper* The remainder of this paper is structured as follows. Our model is introduced in Sect. 2 together with some preliminary results. In Sect. 3, we derive the classical heavy-traffic scaling limits for the queue length process in the presence of overdispersed arrivals both for the moments and the distribution itself. Section 4 presents our main theoretical result, which provides a robust refinement to the heavy-traffic characterization of the queue length measures in pre-limit systems. In Sect. 5, we describe the numerical results and demonstrate the heavy-traffic approximation.

## 2 Model description and preliminaries

*n*, in which time is divided into periods of equal length. At the beginning of each period \(k=1,2,3,\ldots \), new demand \(A_{k,n}\) arrives to the system. The demands per period \(A_{1,n},A_{2,n},\ldots \) are assumed independent and equal in distribution to some nonnegative integer-valued random variable \(A_n\). For brevity, we define \(\mu _n:= {\mathbb {E}}A_{n}\) and \(\sigma _n^2 = \mathrm {Var}\,A_n\). The system has a service capacity \(s_n\in {\mathbb {N}}\) per period, so we have the recursion

*k*steps of a random walk with steps distributed as \(A_n-s_n\). Even more, we can characterize \(Q_{n}\), the stationary queue length, as

### Assumption 1

- (a)(Asymptotic growth)$$\begin{aligned} \mu _n,\sigma _n \rightarrow \infty , \quad \text { for } n\rightarrow \infty . \end{aligned}$$
- (b)(Persistence of overdispersion)$$\begin{aligned} \sigma _n^2/\mu _n \rightarrow \infty , \quad \text { for } n\rightarrow \infty . \end{aligned}$$
- (c)(Heavy-traffic condition) The utilization \(\rho _n := \mu _n/s_n \rightarrow 1\) as \(n\rightarrow \infty \) according tofor some \(\beta > 0\).$$\begin{aligned} (1-\rho _n)\frac{\mu _n}{\sigma _n} \rightarrow \beta , \quad \text {for } n\rightarrow \infty , \end{aligned}$$(4)

*n*grows large. We note that the scenario with \(\sigma _n^2/\mu _n\rightarrow \gamma \) for some \(\gamma > 0\) is asymptotically equivalent to the process studied in [22], in which case overdispersion of the arrival process does not play a role in the limit as \(n\rightarrow \infty \). In order to establish heavy-traffic approximations for large systems that do face overdispersion we need to construct an asymptotic regime in which overdispersion continues to play a dominant role as \(n\rightarrow \infty \), which is secured by Assumption 1(b). The subsequent analysis will clarify why the heavy-traffic condition in Assumption 1(c) is the correct one for our purposes. Note that Assumption 1(c) is satisfied for the capacity allocation rule

*prior*distribution, while \(A_n\) is given the name of a Poisson mixture; see [18]. Given that the moment generation function of \(\Lambda _n\), denoted by \(M^\Lambda _n(\cdot )\), exists, we are able to express the probability generating function (pgf) of \(A_n\) through the former. Namely,

### Lemma 1

*N*(0, 1) denotes a standard normal variable, then \({\hat{A}}_n\) converges weakly to a standard normal variable as \(n\rightarrow \infty \).

The proof can be found in Appendix A.

### Assumption 2

- (a)(Asymptotic regime and persistence of overdispersion)$$\begin{aligned} a_n, b_n \rightarrow \infty , \quad \text { for } n\rightarrow \infty . \end{aligned}$$
- (b)(Heavy-traffic condition) Letfor some \(\beta >0\), or equivalently$$\begin{aligned} s_n = a_n b_n + \beta \sqrt{a_n b_n(b_n+1)} + o\big ( \sqrt{a_n} b_n \big ), \end{aligned}$$$$\begin{aligned} (1-\rho _n)\sqrt{a_n} \rightarrow \beta , \quad \text { for } n\rightarrow \infty . \end{aligned}$$

The next result follows from the fact that \(\Lambda _n\) is a Gamma random variable:

### Corollary 1

Let \(\Lambda _n\sim \text { Gamma}(a_n,1/b_n)\), \(A_n\sim \mathrm{Poisson }(\Lambda _n)\) and \(a_n,b_n\rightarrow \infty \). Then, \({\hat{A}}_n\) converges weakly to a standard normal random variable as \(n\rightarrow \infty \).

### Proof

*G*. By basic properties of the cf,

*t*is in any compact set and

*n*is large enough. By Lévy’s continuity theorem, see, for example, [16, Thm. 18.21], this implies \({\hat{\Lambda }}_n\) is indeed asymptotically standard normal. \(\square \)

The characterization of the arrival process as a Gamma–Poisson mixture is of vital importance in later sections.

### 2.1 Expressions for the stationary distribution

### Assumption 3

The pgf of \(A_n\), denoted by \({\tilde{A}}_n(w)\), exists within \(|w|<r_0\), for some \(r_0>1\), so that all moments of \(A_n\) are finite.

## 3 Heavy-traffic limits

In this section, we present the result on the convergence of the discrete process \({\hat{Q}}_{n}\) to a non-degenerate limiting process and of the associated stationary moments. The latter requires an interchange of limits. Using this asymptotic result, we derive two sets of approximations for \({\mathbb {E}}Q_n\), \(\mathrm{Var}\,Q_n\) and \({\mathbb {P}}(Q_{n}=0)\) that capture the limiting behavior of \(Q_{n}\). The first set provides a rather crude estimation for the first cumulants of the queue length process for any arrival process \(A_{n}\) satisfying Assumption 1. The second set, which is the subject of the next section, is derived for the specific case of a Gamma prior and is therefore expected to provide more accurate, robust approximations for the performance metrics.

We start by indicating how the asymptotic properties of the scaled arrival process give rise to a proper limiting random variable describing the stationary queue length. The asymptotic normality of \({\hat{A}}_{n}\) provides a link with the Gaussian random walk and nearly deterministic queues [36, 37]. The main results in [36, 37] were obtained under the assumption that \(\rho _n\sim 1-\beta /\sqrt{n}\), in which case it follows from [37, Thm. 3] that the rescaled stationary waiting time process converges to a reflected Gaussian random walk.

We shall also identify the Gaussian random walk as the appropriate scaling limit for our stationary system. However, since the normalized natural fluctuations of our system are given by \(\mu _n/\sigma _n\) instead of \(\sqrt{n}\), we assume that the load grows like \(\rho _n \sim 1 - \frac{\beta }{\mu _n/\sigma _n}\). Hence, in contrast to [36, 37], our systems’ characteristics display larger natural fluctuations, due to the mixing factor that drives the arrival process. Yet, by matching this overdispersed demand with the appropriate hedge against variability, we again obtain Gaussian limiting behavior. This is not surprising, since we saw in Lemma 1 that the increments start resembling Gaussian behavior for \(n\rightarrow \infty \). The following result summarizes this.

### Theorem 1

*n*. Then, under Assumption 1, for \(n\rightarrow \infty \),

- (i)
\({\hat{Q}}_{n} {\;\buildrel {d}\over \Rightarrow \;}M_\beta \),

- (ii)
\({\mathbb {P}}(Q_{n} = 0) \rightarrow {\mathbb {P}}(M_\beta =0)\),

- (iii)
\({\mathbb {E}}{\hat{Q}}_{n} \rightarrow {\mathbb {E}}M_\beta \),

- (iv)
\(\mathrm{Var}\,{{\hat{Q}}}_n \rightarrow \mathrm{Var}\,\, M_\beta \),

The proof of Theorem 1 is given in Appendix A. We remark that for convergence of the mean scaled queue length, only \({\mathbb {E}}[(\max \{{\hat{A}}_n,0\})^3]<\infty \) is needed. The following result shows that Theorem 1 also applies to Gamma mixtures, which is a direct consequence of Corollary 1.

### Corollary 2

Let \(\Lambda _n\sim \) Gamma\((a_n,b_n)\). Then, under Assumption 2 the four convergence results of Theorem 1 hold true.

It follows from Theorem 1 that the scaled stationary queueing process converges under (4) to a reflected Gaussian random walk. Hence, the performance measures of the original system should be well approximated by the performance measures of the reflected Gaussian random walk, yielding heavy-traffic approximations.

*x*. Note that this integral involves complex-valued functions with complex arguments. Similar Pollaczek-type integrals exist for \({\mathbb {P}}(M_\beta =0)\) and \(\mathrm{Var}\,M_\beta \); see [1]. The following result simply rewrites these integrals in terms of real integrals and uses the fact that the scaled queue length process mimics the maximum of the Gaussian random walk for large

*n*.

### Corollary 3

### Proof

*x*is any fixed number between 0 and \(2\beta \). Take \(x=\beta \), so that

## 4 Robust heavy-traffic approximations

We shall now establish robust heavy-traffic approximations for the canonical case of Gamma–Poisson mixtures; see (11).

### Theorem 2

The proof of Theorem 2 requires asymptotic evaluation of the Pollaczek-type integrals (20)–(22), for which we shall use a *nonstandard* saddle point method. The saddle point method in its standard form is typically suitable for large deviation regimes, for instance excess probabilities, and it cannot be applied to asymptotically characterize other stationary measures such as the mean or mass at zero. Indeed, in the presence of overdispersion, the saddle point converges to one (as \(n\rightarrow \infty \)), which is a singular point of the integrand, and renders the standard saddle point method useless. Our nonstandard saddle point method, originally proposed by [15] and also applied in [22], aims specifically to overcome this challenge. Subsequently, we apply the nonstandard saddle point method to turn these contour integrals into practical approximations. In contrast to the setting of [22], the analyticity radius tends to one in the setting with overdispersion, which is a singular point of the integrand. For the proof of Theorem 2, we therefore modify the special saddle point method developed in [22] to account for this circumstance.

### Proof

*z*of \(g'(z)\) with \(z\in (1,r_0)\). Since

*z*(

*v*) satisfies

*n*, whereas these bounds \(\pm \tfrac{1}{2} \delta _n\) remained bounded away from zero in [22]. This severely complicates the present analysis. We consider the approximate representation

*O*-term is small compared to \((1-\rho _n)/b_n\) when \(b_n\rightarrow \infty \). Next, we approximate \(r_0\), using that \(r_0>1\) satisfies

*g*(

*z*), as given by (30)–(32), by

We next aim at showing that we have a power series for *z*(*v*) as in (36) that converges for \(|v|\le \tfrac{1}{2}\delta _n\), with \(\tfrac{1}{2}\delta _n\) of the order \(1/b_n\).

### Lemma 2

### Proof

*G*(

*z*) is positive and real for real

*z*close to \(z_{\mathrm{sp}}\). We therefore just need to estimate the convergence radius of this series from below.

*v*with \(|v|\le m_n\), there is exactly one solution \(z=z(v)\) of the equation \(F(z)-iv=0\) in \(|z-z_{\mathrm{sp}}|\le r_n\) by Rouché’s theorem [2]. This

*z*(

*v*) is given by

*v*, \(|v|\le m_n\). From \(|z(v)-z_\mathrm{sp}|\le r_n\), we can finally bound the power series coefficients \(c_k\) according to

### Remark 1

*O*-term in (48) tends to 0 by our assumption that \((1-\rho _n)^2a_n\) is bounded. Thus, we get for \(\mu _{Q_{n}}\) in leading order

## 5 Main insights and numerics

- (i)
At the process level, the space should be normalized with \(\sigma _n\), as in (8). The approximation (27) suggests that it is better to normalize with \(\tilde{\sigma }_n\). Although \(\tilde{\sigma }_n / \sigma _n \rightarrow 1\) for \(n\rightarrow \infty \), the \({\tilde{\sigma }}_n\) is expected to lead to sharper approximations for finite

*n*. - (ii)
Again at the process level, it seems better to replace the original hedge \(\beta \) by the robust hedge \(\beta _n\). This thus means that the original system for finite

*n*is approximated by a Gaussian random walk with drift \(-\beta _n\). Apart from this approximation being asymptotically correct for \(n\rightarrow \infty \), it is also expected to approximate the behavior better for finite*n*.

### 5.1 Convergence of the robust hedge

We next examine the accuracy of the heavy-traffic approximations for \({\mathbb {E}}{Q}_n\) and \(\sigma ^2_Q\), following Corollary 3 and Theorem 2. We expect the robust approximation to be considerably better than the classical approximation when \(\beta _n\) and \(\tilde{\sigma }_n\) differ substantially from their limiting counterparts. Before substantiating this claim numerically, we present a result on the convergence rates of \(\beta _n\) to \(\beta \) and \(\tilde{\sigma }_n\) to \(\sigma _n\).

### Proposition 1

### Proof

Note that \(\beta _n\) always approaches \(\beta \) from below. Also, (51) shows that \(b_n\) is the dominant factor in determining the rate of convergence of \(\beta _n\).

### Proposition 2

### Proof

*n*grows, the substitution of \(\sigma _n\) by \(\tilde{\sigma }_n\) is essential for obtaining accurate approximations, as we illustrate further in the next subsection.

Numerical results for the Gamma–Poisson case with \(\beta =1\) and \(\delta =0.6\)

\(s_n\) | \(\rho _n\) | \({\mathbb {E}}Q_n\) | (25) | (27) | \(\sqrt{\mathrm{Var}\,Q_n}\) | (26) | (29) |
---|---|---|---|---|---|---|---|

5 | 0.609 | 0.343 | 0.246 | 0.363 | 1.002 | 0.835 | 0.978 |

10 | 0.683 | 0.535 | 0.400 | 0.551 | 1.239 | 1.063 | 1.216 |

50 | 0.815 | 1.405 | 1.168 | 1.405 | 1.995 | 1.817 | 1.971 |

100 | 0.855 | 2.113 | 1.824 | 2.105 | 2.445 | 2.270 | 2.420 |

500 | 0.920 | 5.446 | 5.006 | 5.412 | 3.923 | 3.762 | 3.899 |

### 5.2 Comparison between heavy-traffic approximations

Numerical results for the Gamma–Poisson case with \(\beta =1\) and \(\delta =0.8\)

\(s_n\) | \(\rho _n\) | \({\mathbb {E}}Q_n\) | (25) | (27) | \(\sqrt{\mathrm{Var}\,Q_n}\) | (26) | (29) |
---|---|---|---|---|---|---|---|

5 | 0.550 | 0.462 | 0.284 | 0.479 | 1.162 | 0.896 | 1.130 |

10 | 0.587 | 0.852 | 0.521 | 0.855 | 1.570 | 1.213 | 1.528 |

50 | 0.668 | 3.197 | 2.093 | 3.106 | 3.025 | 2.433 | 2.947 |

100 | 0.700 | 5.561 | 3.784 | 5.377 | 3.983 | 3.270 | 3.887 |

500 | 0.766 | 19.887 | 14.741 | 19.202 | 7.514 | 6.455 | 7.361 |

Numerical results for the Gamma–Poisson case with \(\beta =0.1\) and \(\delta =0.6\)

\(s_n\) | \(\rho _n\) | \({\mathbb {E}}Q_n\) | (25) | (27) | \(\sqrt{\mathrm{Var}\,Q_n}\) | (26) | (29) |
---|---|---|---|---|---|---|---|

5 | 0.949 | 11.532 | 11.306 | 11.495 | 3.634 | 3.559 | 3.602 |

10 | 0.961 | 17.565 | 17.268 | 17.548 | 4.474 | 4.398 | 4.444 |

50 | 0.979 | 46.368 | 45.869 | 46.418 | 7.241 | 7.168 | 7.218 |

100 | 0.984 | 70.340 | 69.735 | 70.430 | 8.910 | 8.839 | 8.888 |

500 | 0.991 | 184.900 | 183.989 | 185.108 | 14.422 | 14.357 | 14.404 |

Numerical results for the Gamma–Poisson case with \(\beta =0.1\) and \(\delta =0.8\)

\(s_n\) | \(\rho _n\) | \({\mathbb {E}}Q_n\) | (25) | (27) | \(\sqrt{\mathrm{Var}\,Q_n}\) | (26) | (29) |
---|---|---|---|---|---|---|---|

5 | 0.931 | 15.730 | 15.209 | 15.909 | 4.276 | 4.127 | 4.233 |

10 | 0.939 | 27.561 | 26.672 | 27.958 | 5.652 | 5.466 | 5.605 |

50 | 0.955 | 100.660 | 97.967 | 102.070 | 10.760 | 10.476 | 10.698 |

100 | 0.961 | 175.591 | 171.360 | 177.818 | 14.189 | 13.855 | 14.117 |

500 | 0.971 | 638.097 | 626.346 | 644.105 | 26.963 | 26.490 | 26.864 |

## Notes

### Acknowledgements

The authors are grateful to Avi Mandelbaum for many inspiring discussions and comments. The research of BM was supported by NWO Free Competition Grant No. 613.001.213. The research of JvL is supported by an ERC Starting Grant and by NWO Gravitation Networks Grant No. 024.002.003. The research of BZ is supported by NWO VICI Grant No. 639.033.413.

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