Queues Driven by Hawkes Processes

1 Introduction

• Hawkes arrivals model clustering by making an event increase the likelihood of another soon after, addressing settings where arrivals influence one another.The paper motivates this with socially informed choices and cascading events such as flight cancellations.Source: §1 · pp. 2–3

1.1 Main Contributions of Paper

• The analysis gives exact transient and steady-state queue moments and generating-function results for phase-type service, plus mean, variance, and auto-covariance results for deterministic service.The applications examine web traffic and the impact of a click, and formulate an optimal admission-rate control problem for nightclubs.Source: §1 · p. 3, §1.1 · p. 4

1.2 Organization of Paper

• The paper reviews relevant Hawkes-process results, analyzes deterministic and phase-type service, then applies the results to web traffic and nightclubs.A terminology table defines the paper’s notation, including v for the vector of all ones.Source: §1.2 · pp. 4–5

2 Hawkes Arrival Process

• The Hawkes process couples arrivals to an intensity that jumps by α at each arrival and decays exponentially at rate β toward baseline λ*; this paper assumes an exponential kernel.The process is stable when α < β.Source: §2 · pp. 5–6

2.1 Comparison to the Poisson Process

• When α=0 and λ0=λ*, Hawkes reduces to stationary Poisson; with excitation, arrivals cluster and simulated limit distributions have heavier tails and greater variability.The Hawkes and Poisson processes have the same mean in the limit-distribution comparison.Source: §2.1 · pp. 6–7

2.2 Review of Relevant Hawkes Process Literature

• The Hawkes-process moment equations form a linear ODE system with explicit solutions, and the paper reviews transient and steady-state moments under α < β.The reviewed results include moments of the count and intensity and their covariance.Source: §2.2 · pp. 7–8
• For α ≥ β, the transient mean intensity and count diverge as t→∞, while the queue analysis also considers mean behavior under these unstable arrivals.The paper assumes α < β for most of its analysis.Source: §2.2 · pp. 9–10

2.3 Hawkes/D/∞Queue

• The Hawkes/D/∞ queue is represented as N_t − N_{t−D}, yielding exact transient and steady-state means, transient variance, and transient auto-covariance.The variance and auto-covariance expressions depend on time regimes relative to the deterministic service duration D.Source: §2.3 · pp. 10–12

3.1 Model Definitions and Technical Lemmas

• Phase-type service represents completion through independent exponential sub-services, supports a Markovian queueing model, and can approximate any non-negative continuous distribution.The model starts empty, has infinitely many servers, and tracks customers across n service phases; the subsection also develops generator-based and matrix-integral lemmas for the analysis.Source: §3.1 · pp. 13, 14, 16

3.2 Mean Dynamics of the Hawkes/PH/∞Queue

• The authors derive differential equations for moments of phase counts and intensity, with closed linear systems for moments up to any fixed order.At order two, the system gives equations for means, variances, and covariances of counts in each phase and in the queue as a whole.Source: §3.2 · pp. 18, 20, 21
• Corollary 3.8 gives differential equations for phase-level means, variances, and covariances, including covariances between the Hawkes intensity and queue counts.The covariance equations cover both pairs of service phases and the intensity–phase-count relationship.Source: §3.2 · pp. 20–21
• For α < β and invertible S + (β − α)I, the analysis gives exact expressions for phase-specific means, intensity–queue covariances, and covariances between service phases.The stated invertibility condition is not necessary for a closed-form solution; the paper also treats Erlang and hyper-exponential service when it fails.Source: §3.2 · pp. 22, 25
• For Erlang and hyper-exponential service, the authors derive explicit transient phase means and covariances, including cases where a service rate matches β − α.The results cover phase counts, their covariances with the intensity, and covariances between phases.Source: §3.2 · pp. 25–27

3.3 Limiting Behavior of the Hawkes/PH/∞Queue

• In the stable regime, steady-state phase means and covariances are derived, with phase covariances given by a Lyapunov equation; for exponential service, variance differs from the mean by a service-duration-scaled queue–intensity covariance.The latter relationship contrasts with the Poisson-driven M/M/∞ queue, whose steady-state variance equals its mean.Source: §3.3 · pp. 29–30
• When α ≥ β, steady-state analysis does not apply, and the authors instead derive the transient mean number in each phase for phase-type service.The case α = β is separately identified in the result.Source: §3.3 · p. 30

3.4 Auto-covariance of the Hawkes/PH/∞Queue

• For stable Hawkes arrivals and exponential service, the authors give an explicit queue auto-covariance for t ≥ τ, including special cases when service-rate expressions coincide with Hawkes decay rates.The formula treats the matching-rate case µ = β − α and defines a separate function h(s) according to whether 2µ = β − α.Source: §3.4 · pp. 31–32

3.5 Generating Functions for the Hawkes/PH/∞Queue

• For phase-type service, the queue moment generating function and its cumulant counterpart are characterized by partial differential equations.The cumulant generating function PDE depends on partial derivatives rather than on the function itself.Source: §3.5 · pp. 34–35
• The cumulant moment generating function also has a representation using an ordinary differential equation, obtained through the method of characteristics.The theorem defines h(z) as the solution to that ODE.Source: §3.5 · p. 36
• Reducing the cumulant-generating-function PDE to an ODE simplifies numerical implementation; the moment generating function follows by exponentiating the cumulant generating function.The ODE may lack a closed-form solution outside special cases.Source: §3.5 · p. 37

3.6 Simulation Study

• Simulations using 100,000 replications verify moment expressions for single-phase, three-phase Erlang, and three-phase hyper-exponential service settings.The examples include means, variances, queue-intensity covariance, and phase or queue covariances.Source: §3.6 · pp. 37–38
• For equal Hawkes parameters and equivalent service parameters, the steady-state variance of the Hawkes/D/∞ queue exceeds that of the Hawkes/M/∞ queue.The simulations also find larger variance with deterministic service; the paper attributes this to clustered departures, which random service durations can disperse.Source: §3.6 · p. 39
• In lognormal-service simulations, queue means converge to 1, higher service variance slows mean convergence, and higher service variance lowers queue variance.The lognormal distributions have mean 1 and variances from 0 to 5 in increments of 0.5; simulations use 10,000 replications.Source: §3.6 · pp. 40–41

4.1 Trending Web Traffic

• In 106 tweets posting Young Jeezy’s “My President” video, 60 arrived within an hour of the 2012 election results, with other posting clusters also observed.Simulations found a majority of arrivals within one time quartile in 82.4% of trials for λ∗=0.5, α=19.5, β=20, versus 18.0% for λ∗=1, α=0.5, β=1.Source: §4.1 · p. 42, Figure 4.1 · p. 43
• The click experiment models a click as an extra arrival in an otherwise identical Hawkes process and compares expected dwell-time revenue with the process starting empty.The cited results state that the revenue gaps grow with time and tend to infinity as α approaches β.Source: §4.1 · pp. 43–45

4.2 Club Queue

• The Club Queue models outside admission and time inside the nightclub as two service phases, with an optimal admission rate balancing revenue against queue-length and admission-rate penalties.The objective is considered over a finite horizon [0, T].Source: §4.2 · pp. 46–47
• Changing the penalty and target admission rate substantially changes the optimal control: one scenario allows an outside queue roughly three times larger, while the other has an admission rate about twice as high.The scenarios differ in w and k; the reported comparison does not isolate either parameter’s effect.Source: §4.2 · p. 48

• The analysis derives exact moments and a moment generating function for a Hawkes-driven infinite-server queue with phase-type service.The authors also describe the queueing model as analyzed in detail.Source: §5 · p. 48
• A non-stationary baseline intensity remains a future extension; separating time-varying arrival rates from self-excitation is an important motivation for studying it.A periodic baseline such as λ*(t) = λ* + ρ · sin(t) is described as seemingly analytically tractable, while other functions may require Fourier analysis.Source: §5 · p. 48
• Applying the analysis to queues beyond the infinite-server model, including Erlang-A with abandonments, remains future work.The authors describe analytical expressions for Erlang-A as non-trivial, even with Poisson arrivals, and propose closure approximations or spectral expansions for the Hawkes-driven setting.Source: §5 · p. 49

• Proposition 5.1 concerns the Hawkes/PH/∞ queue described in Section 3.Source: §A.1 · p. 53
• The proposition assumes S + (β − α)I is nonsingular and applies for t ≥ τ ≥ 0.Source: §A.1 · p. 53
• The proof obtains the stated result by substituting Theorem 3.9 expressions into Equation 3.33.Source: §A.1 · p. 54