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This is the Exam of Statistical Science which includes Upcrossing Inequality, Understood, Time Series and Monte Carlo Inference Two, Second Order Stationary, Telephone Banking Facility, Headline, Supermarkets etc. Key important points are: Stochastic Networks, Closed Migration Process, Stationary Distribution, Telephone Switchboard, Incoming Lines, Poisson Process, Finding, Free Line, Exponentially Distributed, Distributed Length
Typology: Exams
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Monday 11 June 2001 9 to 11
Attempt any THREE questions. The questions carry equal weight.
1 Define a closed migration process. Establish the form of the stationary distribution of a closed migration process.
A telephone switchboard has N incoming lines and one operator. Calls to the switchboard are initiated as a Poisson process of rate ν, but calls initiated when all N lines are in use are lost. A call finding a free line has then to wait for the operator to answer. The operator deals with waiting calls one at a time, and takes an exponentially distributed length of time with mean λ−^1 to connect a call to the correct extension, after which the call lasts for an exponentially distributed length of time with mean μ−^1. All these lengths of time are independent of each other and of the initiating Poisson process. Model the system as a closed migration process, and show that in equilibrium the proportion of calls lost is
n=
H(n)
where
H(n) =
( (^) ν
λ
)n ∑n
i=
λ μ
)i 1 i!
2 Define the Erlang fixed-point approximation for a loss network with fixed routing, and establish the existence and uniqueness of the approximation.
Show, by means of an example or otherwise, that in a loss network with alternative routing the natural generalization of the Erlang fixed-point approximation may not be unique.
3 Outline a mathematical model of the slotted infinite-population ALOHA random access protocol, obtaining the recurrence
Nt+1 = Nt + Yt− 1 − I[Zt = 1],
where Zt = 0, 1 or ∗ according as 0, 1 or more than 1 packets are transmitted in slot (t, t + 1), and Yt is the number of arrivals in slot (t, t + 1). What does Nt represent?
Prove that for any positive arrival rate
P {∃J < ∞ : Zt = ∗, for all t > J} = 1.
Discuss whether we can expect a similar result for a finite-population model.
