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supposenbspanbspandnbspbnbspare each ergodic markov chains with transition prob- abilities paiajnbsp and pbibjnbsp
section 451 showed how to find the expected first-passage times to a fixed state say 1 from all other states it is
anbspassume throughout that p is the transition matrix of a unichain and thus the eigenvalue 1 has multiplicity 1 show
consider the markov chain belowa suppose the chain is started in statenbspinbspand goes throughnbspnnbsptransitions
consider finding the expected time until a given string appears in a iid binary sequence with prxn 1 p1 prxn 0 p0
consider a markov decision problem with m states in which some state say state 1 is inherently reachable from each
george drives his car to the theater which is at the end of a one-way street there are parking places along the side of
consider a markov decision problem in which the stationary policiesnbspk andnbspklowastnbspeach satisfy 450 and each
the odoni bound let klowast be the optimal stationary policy for a markov decision problem and let glowast and pi
consider an integer-time queueing system with a finite buffer of size 2 at the beginning of thenbspnth time interval
the purpose of this exercise is to show that for an arbitrary renewal process nt the number of renewals in 0 t is a
let xinbspinbspge 1 be the inter-renewal intervals of a renewal process gener- alized to allow for inter-renewal
a town starts a mosquito control program and the rvnbspznnbspis the number of mosquitoes at the end of thenbspnth year
let xinbspige1 be the inter-renewal intervals of a renewal process and assume that e xi infin letnbspb gtnbsp0 be an
consider a variation of an mg1 queueing system in which there is no facility to save waiting customers assume customers
a gambler with an initial finite capital ofnbspdnbsp gtnbspnbsp0 dollars starts to nbspplay a dollar slot machine at
letnbspjnbsp minnnbspnbspsnlebnbspornbspsngea wherenbspanbspis a positive integernbspbnbspis a negative integer
let ntnbspt gtnbsp0 be a renewal counting process generalized to allow for inter-renewal intervals xi of duration 0 let
anbsplet the inter-renewal interval of a renewal process have a second- order erlang density fxnbspx
anbspletnbspnt be the number of arrivals in the interval 0nbspt for a poisson process of ratenbsplambda show that the
a findnbsp limtrarrinfine ntnbsp -nbsptxnbsp fornbsp anbsp renewalnbsp counting process ntnbspt gtnbsp0 with
customers arrive at a bus stop according to a poisson process ofnbsp nbspratenbsplambda independently buses arrive
consider the same setup as in exercise 531 except that now customers arrive according to a non-arithmetic renewal
an mg1 queue has arrivals at ratenbsplambdanbspand a service time distribution given by fynbspy assume
consider a sequencenbspx1nbspx2nbspnbspof iid binary rv s with prxnnbsp 1 nbspp1 and prxnnbsp 0 nbspp0 1 -nbspp1