Infinitesimal Perturbation Analysis in Networks of Stochastic Flow Models: General Framework and Case Study of Tandem Networks with Flow Control

Abstract

This paper presents a general algorithmic framework for computing the IPA derivatives of sample performance functions defined on networks of fluid queues. The underlying network-model consists of bi-layered hybrid dynamical systems with continuous-time dynamics at the lower layer and discrete-event dynamics at the upper layer. The linearized system, computed from the sample path via a discrete-event process, yields fairly simple algorithms for the IPA derivatives. As an application-example, the paper discusses loss and workload performance functions in a tandem network with congestion control, subjected to signal delays.

Appendix

The purpose of this section is to prove Lemma 1 and Proposition 9.

Proof

[Proof of Lemma 1.] Define Z: = {z _i,k,i = 1,2; k = 1,2,...}, namely the set [0,T] ∖ Z consists of the union of the open intervals $I_{i,k}^o$, i = 1,2; k = 1,2,.... The proof is by induction. Define T _min : =  min {T ₁₁,T ₁₂,T ₂₁}. Fix t ∈ [0,T] ∖ Z, and suppose that for every τ ∈ [0,t] ∖ Z, $\frac{\partial\alpha_{i}}{\partial\theta}(\theta,\tau)=0$ and $\frac{\partial\gamma_{i}}{\partial\theta}(\theta,\tau)=0$, for both i = 1,2. We will show that the same equations hold for all τ ∈ [t,t + T _min ] ∖ Z as well, and this will complete the proof.

Consider first the case where i = 1. Fix τ ∈ [t,t + T _min ] ∖ Z. If Q ₁ was not full at time τ − T ₁₁ and Q ₂ was not full at time τ − T ₂₁, then α ₁(θ,τ) = σ(τ), and either γ ₁(θ,τ) = σ(τ) − β ₁(τ) or γ ₁(θ,τ) = 0; hence $\frac{\partial\alpha_{1}}{\partial\theta}(\theta,\tau)=0$ and $\frac{\partial\gamma_{1}}{\partial\theta}(\theta,\tau)=0$. Next, if Q ₁ was not full at time τ − T ₁₁ while Q ₂ was full at time τ − T ₂₁, then α ₁(θ,τ) = σ(τ) − c ₂ γ ₂(θ,τ − T ₂₁). But τ − T ₂₁ ≤ t, and hence, and by the above induction’s hypothesis, $\frac{\partial\alpha_{1}}{\partial\theta}(\theta,\tau)=0$. Moreover, either γ ₁(θ,τ) = α ₁(θ,τ) − β ₁(τ) or γ ₁(θ,τ) = 0, and hence we obtain that $\frac{\partial\gamma_{i}}{\partial\theta}(\theta,\tau)=0$ as well. The other cases, namely where Q ₁ was full at time τ − T ₁₁, yield similar arguments, and also the case of i = 2 is provable in a similar fashion. This completes the proof. □

Remark 5

The above proof of Lemma 1 is based on the assumption, made throughout the paper, that T _min  > 0. If T _min  = 0 then Lemma 1 still holds true, and it admits a simpler proof. The proof is based on the fact that α _i (θ,t) and γ _i (θ,t) can be expressed as simple functions of σ(t), β ₁(t), and β ₂(t), similarly to Lemma 4.1 in Wardi et al. (2009). We do not present the proof here since this paper concerns the case of positive signal delays.

Proof

(Proof of Proposition 9.) We establish the two conditions of Proposition 1. The first condition is satisfied by Assumption 2 and Proposition 2. As for the second condition, it is evident that (w.p.1) the functions J _L,i(θ) and J _W,i(θ) are piecewise continuously differentiable, and hence, it suffices to prove that there exists a random number K > 0 having a finite first moment such that $\sup_{\theta\in\Theta}|\frac{dJ_{L,i}}{d\theta}(\theta)|\leq K$, and $\sup_{\theta\in\Theta}|\frac{dJ_{W,i}}{d\theta}(\theta)|\leq K$.

Since T < ∞ and T _ij  > 0, there exists a constant K ₁ such that the number of induced events at either queue is bounded from above by K ₁. By Assumption 2(i) there exists a random number K ₂ > 0 such that, the number of boundary periods at either queue is bounded from above by K ₂. Finally, by Proposition 7, in conjunction with Propositions 5 and 6, and Eq. 22, it follows that for all θ ∈ Θ, $|\frac{dJ_{L,i}}{d\theta}(\theta)|\leq 2K_{1}K_{2}$. Similarly, by Proposition 8, $|\frac{dJ_{W,i}}{d\theta}(\theta)|\leq 2K_{1}K_{2}T$. This completes the proof. □