Modeling and Optimization of 𝑀/𝐺/𝟏-Type Queueing Networks: An Efficient Sensitivity Analysis Approach

Table 1

Notations.

Symbol	Definition
$𝒢=(𝒱,ℰ)$	Graph representation of the network
$C_{\iota }$	Service capacity of link $\iota$
$N(t)$	Number of active user applications served within a network element
$C_o(t)$	Leftover service capacity
$G_i$	General distribution of the $i$th user queue's interservice time
${\lambda }_i$	The long-term average rate of $i$th user arrivals
${\mu }_i$	Mean service rates of the $i$th user queue
$K$	Each user queue's backlog capacity
$X^i$	Semi-Markov process of each $M/G/1$-type user queue
$\mathrm{\Phi }$	The state space of each user queue
$Y^i$	The embedded Markov chain of the semi-Markov process
$Q^{{\mu }_i}(t)$	The semi-Markov kernel
$P^{{\mu }_i}$	The transition probability matrix
$A^{{\mu }_i}$	The infinitesimal generator
${\pi }^{{\mu }_i}$	The steady-state probability vector
$f^{{\mu }_i}$ ($f_i$)	The performance cost function
${\eta }_f^{{\mu }_i}$	The performance cost measure
$g^{{\mu }_i}$	The performance potential vector
$D_f^{{\mu }_i}$	The realization matrix
${\mathrm{\Gamma }}_i$	The feasible region of the service rates allocated to the $i$th user queue
$\mathrm{\Omega }(t)$	The policy space for all user queues
$\nu (t)$	A feasible resource allocation policy
$ϵ$	The stopping criterion of the policy algorithm
$l$	The iteration index of the policy algorithm
$\text{sp}$	The span seminorm