Mean Value Analysis of Closed G-Networks with Signals

Abstract

We consider a closed network of queues with external signals. These signals trigger customer between queues and they arrive following a rate which depends on the number of active customers in the station. We consider three types of stations: they may have one server, an infinite number of servers or no servers at all. In that case, the customers behave like inert customers and they only react to signal. We prove that, under irreducibility conditions, such a network has a stationary distribution which is multiplicative. As the network is finite, all the states are not reachable and the distribution is known up to a normalization constant. To avoid the computation of this constant, we also prove a mean value analysis algorithm which allows to determine the average queue size and the average waiting time without computing the probabilities. We also present some extensions of the model.

Appendix: Proof of Theorem 1

Consider again the global balance equation at steady-state.

$$\begin{aligned} \pi (K,\varvec{x})[\sum \nolimits _{i \in {\mathcal{{F}}}}&\mu _i 1_{x_i>0} + \sum \nolimits _{i \in {\mathcal{{I}}}} \mu _i x_i + \lambda ^t (\sum \nolimits _{i \in {\mathcal{{F}}}} \frac{1_{x_i>0}}{K} + \sum \nolimits _{i \in {\mathcal{{I}}}\cup {\mathcal{{Z}}}} \frac{x_i}{K} )]&\\&= \sum \nolimits _{i \in {\mathcal{{F}}}} \sum \nolimits _{j \in {\mathcal{{F}}}\cup {\mathcal{{I}}}\cup {\mathcal{{Z}}}} \mu _i \pi (K,\varvec{x} +\varvec{e_i} -\varvec{e_j}) \mathbf{R} [i,j] 1_{x_j>0}&[1] \\&+ \sum \nolimits _{i \in {\mathcal{{I}}}} \sum \nolimits _{j \in {\mathcal{{F}}}\cup {\mathcal{{I}}}\cup {\mathcal{{Z}}}} x_i \mu _i \pi (K,\varvec{x} +\varvec{e_i} -\varvec{e_j}) \mathbf{R} [i,j] 1_{x_j>0}&[2] \\&+ \sum \nolimits _{i \in {\mathcal{{F}}}} \sum \nolimits _{j \in {\mathcal{{F}}}\cup {\mathcal{{I}}}\cup {\mathcal{{Z}}}} \pi (K,\varvec{x} +\varvec{e_i} -\varvec{e_j}) \mathbf{T} [i,j] 1_{x_j>0} \lambda ^t \frac{1_{x_i + 1>0}}{K}&[3] \\&+ \sum \nolimits _{i \in {\mathcal{{I}}}} \sum \nolimits _{j \in {\mathcal{{F}}}\cup {\mathcal{{I}}}\cup {\mathcal{{Z}}}} \pi (K,\varvec{x} +\varvec{e_i} -\varvec{e_j}) \mathbf{T} [i,j] 1_{x_j>0} \lambda ^t \frac{ {x_i + 1 }}{K}&[4] \\&+ \sum \nolimits _{i \in {\mathcal{{Z}}}} \sum \nolimits _{j \in {\mathcal{{F}}}\cup {\mathcal{{I}}}\cup {\mathcal{{Z}}}} \pi (K,\varvec{x} +\varvec{e_i} -\varvec{e_j}) \mathbf{T} [i,j] 1_{x_j>0} \lambda ^t \frac{ {x_i + 1 }}{K}&[5]\\ \end{aligned}$$

Divide both sides by $\pi (\varvec{x})$ and take into account the multiplicative solution proposed in Eq. 1. As the probability depends on the type of station, we have to decompose the summation into three parts, based on the set of stations we consider. We also notice that $1_{x_i + 1 >0} =1$ and that $x_i 1_{x_i>0}= x_i$ and we simplify some terms.

$$\begin{aligned} \sum \nolimits _{i \in {\mathcal{{F}}}} \mu _i 1_{x_i>0}&+ \sum \nolimits _{i \in {\mathcal{{I}}}} \mu _i x_i + \lambda ^t (\sum _{i \in {\mathcal{{F}}}} \frac{1_{x_i>0}}{K} + \sum \nolimits _{i \in {\mathcal{{I}}}\cup {\mathcal{{Z}}}} \frac{x_i}{K} )&~&\\&= \sum \nolimits _{i \in {\mathcal{{F}}}} \sum \nolimits _{j \in {\mathcal{{F}}}} \mu _i \rho _i/\rho _j \mathbf{R} [i,j] 1_{x_j>0}&[1] \\&+ \sum \nolimits _{i \in {\mathcal{{F}}}} \sum \nolimits _{j \in {\mathcal{{I}}}} \mu _i \rho _i/\rho _j x_j \mathbf{R} [i,j]&[2] \\&+ \sum \nolimits _{i \in {\mathcal{{F}}}} \sum \nolimits _{j \in {\mathcal{{Z}}}} \mu _i \rho _i/\gamma _j x_j \mathbf{R} [i,j]&[3] \\&+ \sum \nolimits _{i \in {\mathcal{{I}}}} \sum \nolimits _{j \in {\mathcal{{F}}}} \mu _i \rho _i/\rho _j \mathbf{R} [i,j] 1_{x_j>0}&[4] \\&+ \sum \nolimits _{i \in {\mathcal{{I}}}} \sum \nolimits _{j \in {\mathcal{{I}}}} \mu _i \rho _i/\rho _j x_j \mathbf{R} [i,j]&[5] \\&+ \sum \nolimits _{i \in {\mathcal{{I}}}} \sum \nolimits _{j \in {\mathcal{{Z}}}} \mu _i \rho _i/\gamma _j x_j \mathbf{R} [i,j]&[6] \\&+ \sum \nolimits _{i \in {\mathcal{{F}}}} \sum \nolimits _{j \in {\mathcal{{F}}}} \rho _i/\rho _j \mathbf{T} [i,j] 1_{x_j>0} \lambda ^t \frac{1}{K}&[7] \\&+ \sum \nolimits _{i \in {\mathcal{{F}}}} \sum \nolimits _{j \in {\mathcal{{I}}}} \rho _i/\rho _j x_j \mathbf{T} [i,j] \lambda ^t \frac{1}{K}&[8] \\&+ \sum \nolimits _{i \in {\mathcal{{F}}}} \sum \nolimits _{j \in {\mathcal{{Z}}}} \rho _i/\gamma _j x_j \mathbf{T} [i,j] \lambda ^t \frac{1}{K}&[9] \\&+ \sum \nolimits _{i \in {\mathcal{{I}}}} \sum \nolimits _{j \in {\mathcal{{F}}}} \rho _i/\rho _j \mathbf {T(i,j)} \mathbf{T} [i,j] \lambda ^t \frac{1}{K}&[10] \\&+ \sum \nolimits _{i \in {\mathcal{{I}}}} \sum \nolimits _{j \in {\mathcal{{I}}}} \rho _i/\rho _j x_j \mathbf{T} [i,j] \lambda ^t \frac{ 1 }{K}&[11] \\&+ \sum \nolimits _{i \in {\mathcal{{I}}}} \sum \nolimits _{j \in {\mathcal{{Z}}}} \rho _i/\gamma _j x_j \mathbf{T} [i,j] \lambda ^t \frac{ 1 }{K}&[12] \\&+ \sum \nolimits _{i \in {\mathcal{{Z}}}} \sum \nolimits _{j \in {\mathcal{{F}}}} \gamma _i/\rho _j \mathbf{T} [i,j] 1_{x_j>0} \lambda ^t \frac{ 1 }{K}&[13]\\&+ \sum \nolimits _{i \in {\mathcal{{Z}}}} \sum \nolimits _{j \in {\mathcal{{I}}}} \gamma _i/\rho _j x_j \mathbf{T} [i,j] \lambda ^t \frac{ 1 }{K}&[14]\\&+ \sum \nolimits _{i \in {\mathcal{{Z}}}} \sum \nolimits _{j \in {\mathcal{{Z}}}} \gamma _i/\gamma _j x_j \mathbf{T} [i,j] \lambda ^t \frac{ 1 }{K}&[15]\\ \end{aligned}$$

We exchange the role of indices i and j in all the terms of the r.h.s. and we factorize the terms ($1+4+7+10+13$), ($2+5+8+11+14$), ($3+6+9+12+15$):

$$\begin{aligned} \sum \nolimits _{i \in {\mathcal{{F}}}} \mu _i 1_{x_i>0}&+ \sum \nolimits _{i \in {\mathcal{{I}}}} \mu _i x_i + \lambda ^t (\sum \nolimits _{i \in {\mathcal{{F}}}} \frac{1_{x_i>0}}{K} + \sum \nolimits _{i \in {\mathcal{{I}}}\cup {\mathcal{{Z}}}} \frac{x_i}{K} )&\\ \\&= \sum \nolimits _{i \in {\mathcal{{F}}}} 1_{x_i>0} 1/\rho _i \left[ \sum \nolimits _{j \in {\mathcal{{F}}}\cup {\mathcal{{I}}}} \mu _j \rho _j \mathbf{R} [j,i]~+~\sum \nolimits _{j \in {\mathcal{{F}}}} \rho _j \mathbf{T} [j,i] \frac{\lambda ^t}{K} \right.&\\ \\&\qquad \qquad \qquad \qquad \quad \left. +\,\sum \nolimits _{j \in {\mathcal{{I}}}} \rho _j \mathbf{T} [j,i] \frac{\lambda ^t}{K}~+~\sum \nolimits _{j \in {\mathcal{{Z}}}} \gamma _j \mathbf{T} [j,i] \frac{\lambda ^t}{K} \right]&\\ \\&+ \sum \nolimits _{i \in {\mathcal{{I}}}} x_i/\rho _i \left[ \sum \nolimits _{j \in {\mathcal{{F}}}\cup {\mathcal{{I}}}} \mu _j \rho _j \mathbf{R} [j,i]~+~\sum \nolimits _{j \in {\mathcal{{F}}}} \rho _j \mathbf{T} [j,i] \frac{\lambda ^t}{K} \right.&\\ \\&\qquad \qquad \qquad \quad \left. +\,\sum \nolimits _{j \in {\mathcal{{I}}}} \rho _j \mathbf{T} [j,i] \frac{\lambda ^t}{K}~+~\sum \nolimits _{j \in {\mathcal{{Z}}}} \gamma _j \mathbf{T} [j,i] \frac{\lambda ^t}{K} \right]&\\&\\&+\sum \nolimits _{i \in {\mathcal{{Z}}}} x_i/\gamma _i \left[ \sum \nolimits _{j \in {\mathcal{{F}}}\cup {\mathcal{{I}}}} \mu _j \rho _j \mathbf{R} [j,i]~+~\sum \nolimits _{j \in {\mathcal{{F}}}} \rho _j \mathbf{T} [j,i] \frac{ \lambda ^t }{K} \right.&\\ \\&\qquad \qquad \qquad \quad \left. \sum \nolimits _{j \in {\mathcal{{I}}}} \rho _j \mathbf{T} [j,i] \frac{\lambda ^t}{K}~+~\sum \nolimits _{j \in {\mathcal{{Z}}}} \gamma _j \mathbf{T} [j,i] \frac{\lambda ^t}{K}\right] .&\\&\\ \end{aligned}$$

The first and second term of the l.h.s. cancel with the first and second term of the r.h.s. due to the first flow equation (i.e. Eq. 2). It remains:

$$ \begin{array} {rcll} \lambda ^t \sum \nolimits _{i \in {\mathcal{{Z}}}} \frac{x_i}{K} &{} = &{}\sum \nolimits _{i \in {\mathcal{{Z}}}} x_i/\gamma _i \left[ \sum \nolimits _{j \in {\mathcal{{F}}}\cup {\mathcal{{I}}}} \mu _j \rho _j \mathbf{R} [j,i]~+~\sum \nolimits _{j \in {\mathcal{{F}}}} \rho _j \mathbf{T} [j,i] \frac{ \lambda ^t }{K} \right. &{} \\ \\ &{}&{}~\left. \sum \nolimits _{j \in {\mathcal{{I}}}} \rho _j \mathbf{T}[j,i] \frac{\lambda ^t}{K}~+~\sum \nolimits _{j \in {\mathcal{{Z}}}} \gamma _j \mathbf{T}[j,i] \frac{\lambda ^t}{K}\right] . &{} \\ &{}&{}&{} \\ \end{array} $$

And this last equation is equivalent to the second flow equation for station in ${\mathcal{{Z}}}$ (i.e. Eq. 3). And the proof is complete.    $\square$