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For route flows $x\ge 0$, let the link-route incidence matrix $A$ give link loads $y=Ax$. Let the source-sink route incidence matrix $H$ enforce fixed demands $Hx=f$. With continuous nondecreasing link delays $D_j$, the Beckmann potential isIts route derivative is ${\partial }_{r}V={\sum }_j{A}_{jr}{D}_j(y_j)$, precisely that route's delay. Convexity follows from monotonicity of each $D_j$. The Karush-Kuhn-Tucker conditions therefore say that used routes have the minimum delay for their source-sink pair, while unused routes have no smaller delay: the Wardrop equilibrium conditions.^[ref]

This potential generally differs from total travel delay ${\sum }_j{y}_j{D}_j(y_j)$. For two parallel links with delays $D_1(y)=y$, $D_2(y)=1$ and total demand two, minimizing $V=x_1^2/2+x_2$ gives $x_1=x_2=1$. Both used routes have delay one. Minimizing total delay $x_1^2+x_2$ instead gives $x_1=1/2$, $x_2=3/2$. The marginal external cost toll aligns the two optimization problems.

Elena Yudovina is a mathematician whose work includes the theory of stochastic networks. With Frank Kelly, she wrote Stochastic Networks by Frank Kelly and Elena Yudovina, connecting queueing theory, loss networks, congestion control and distributed optimization. The author's book page verifies the joint authorship and 2014 publication.^[ref]

A central question is how local service or routing rules affect an entire network. The book's M/M/1 queue illustrates the danger of operating near capacity: with arrival rate $\lambda$, service rate $\mu$ and $\rho =\lambda /\mu <1$, mean queue population is $\rho /(1-\rho )$. Modest changes in load can therefore cause large changes in delay.^[ref]

The Erlang loss formula gives the blocking probability for a single resource with $C$ identical circuits, no waiting room, independent Poisson arrivals of rate $\lambda$ and independent exponential holding times of mean $1/\mu$. Each admitted call occupies one circuit. Write offered traffic $a=\lambda /\mu$.

The occupied-circuit count is a birth-death process on $0,\dots ,C$, with upward rate $\lambda$ below capacity and downward rate $n\mu$. Detailed balance gives $\lambda \pi (n)=(n+1)\mu \pi (n+1)$, hence $\pi (n)\propto a^n/n!$. By Poisson arrivals see time averages, the arrival blocking probability isThe resource has a finite state space even when $a>C$; excess calls are rejected rather than accumulating.^[ref]

For $C=2$, $a=1$, the weights are $1,1,1/2$, giving blocking $1/5$ and carried load of an Erlang loss resource $a[1-E(a,C)]=4/5$. A useful recursion avoids large factorials:The insensitivity of loss networks extends the occupancy formula to independent general holding times with the same mean under the standard loss-model assumptions.

Frank Kelly works on probability theory, stochastic networks and the mathematical organization of shared resources. His research connects queueing theory with optimization: local decisions about service, admission or price can determine a network's aggregate performance. His institutional research record describes these interests.^[ref]

With Elena Yudovina, Kelly wrote Stochastic Networks by Frank Kelly and Elena Yudovina. Its treatment of loss networks shows how detailed balance can produce an explicit equilibrium occupancy law, while congestion control uses resource prices to coordinate competing demands. These are complementary uses of mathematics: predicting congestion and designing rules that improve it.^[ref]

Let $N_t$ be the backlog in slotted ALOHA, and let stations use attempt probability $1/S_t$ with estimate $S_t\ge 1$. Update the estimate by $a$ after an idle slot, $b$ after a success and $c$ after a collision, clipping its value below at one. These feedback parameters therefore specify an actual controller, rather than only a drift formula.^[ref]

In the interior fluid model, write $s,n>0$ and $\kappa =n/s$. The idle and success probabilities approach $e^{-\kappa }$ and $\kappa e^{-\kappa }$. For $a=b=-A$, $c=2A/(e-2)$ and $A\ge 1$, their weighted mean givesHere $\nu$ is the mean number of arrivals per slot. Since $g(1)=0$ and $g^{\prime }(\kappa )=Ae\kappa e^{-\kappa }/(e-2)>0$, the estimate falls below contention ratio one and rises above it.

Use the ratio time change for a homogeneous fluid model. Its ratio drift is $h=\nu -\kappa e^{-\kappa }-\kappa g$. For $\kappa \ge 1$,because $g\ge 0$ and $Ae{\kappa }^2/(e-2)>\kappa -1$. If $\nu <e^{-1}$, then $h(1)<0$; continuity supplies a trapping upper bound below one. There $g$ is bounded strictly below zero, giving finite-time draining of a fluid model. If $\nu >e^{-1}$, instead $\dot{n}\ge \nu -e^{-1}>0$. These are fluid conclusions; stochastic stability requires a separate argument.