Lecture 3 Notes: Geometrical Probability in the Sky (Part 1)
Suppose that two aerial routes-one Eastbound and one Northbound-cross at an
altitude of 35,000 feet at junction J (Figure 1). In the absence of air-traffic control, the
times at which eastbound

Lecture 6 Notes: The M/G/1 Queueing System (Part
1)
For the M/G/1 queueing system being operated under the FIFO service rule, we derive the expressions ofthe of llowing quantities in terms ofthe arrival rate , the mean service time E[S], and
the variance

Lecture 4 Notes: Geometrical Probability in the Sky (Part 2)
P(N):
The reasoning is the same as for P(E), so we can write:
P(N) = 1 - exp(- N)
It might seem surprising that P(E) and P(N) differ, given that each conflict we are
considering involves one eas

Lecture 5 Notes: Transportation
Models
Suppose that emergency vehicles are distributed over a region under a spatial
Poisson process with parameter . Let vi be the straight-line distance from point P to
the ith nearest emergency vehicle. We seek P(v2 > 2v

Lecture 2 Notes: Croftons Method
Let X1 and X2 be independent random variables that are uniformly distributed over the
interval [0, a]. We are interested in computing E[|X1 X2 |]. For instance, in an urban setting,
X1 and X2 may denote the location of an

Lecture 1 Notes: Wall of
Shame Problem
Wall of Shame
Assume a square city of side L, with the borders going NorthSouth and
EastWest. A malicious mayor decides to create an East West barrier a distance x north
of the citys southern border, which will have

Lecture 7 Notes: The M/G/1 Queueing System (Part
2)
Note that
n1
nPn = L and
n1
Pn = . So we have
E[T2 ] = E[S]L E[S] .
Now let us compute E[T1 ]. Since the service time distribution may not be negative exponential,
we should consider the issue of random

Lecture 10 Notes: Preemptive Priority Queues (Part 3)
The time it will take the server to clear the system of all k2 of
the Type-2s already present is simply k2 E2(s); it will take her an
additional E2(s) to serve the new arrival.
It follows that:
E2(wk1,

Lecture 8 Notes: PREEMPTIVE PRIORITY QUEUES (Part 1)
Consider an M/M/1 queuing system in which there are two
classes of customers-high and low priority-who arrive under
independent Poisson processes with parameters of, respectively, 1
and 2. We assume tha

Lecture 9 Notes: PREEMPTIVE PRIORITY QUEUES (Part 2)
The Type-2 Ordeal
A Type-2 customers service can be humiliating and
protracted. He starts service and either (i) finishes up before
any Type-1s arrive, or(ii) is instead preempted by a Type-1
arrival. O