# Class i connections in progress 42 47 birth processes

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Unformatted text preview: th-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Call blocking in an ATM network An ATM network offers calls of two different types. R1 = 1Mbps λ = arrival rate 1 µ1 = mean holding time R2 = 2Mbps λ = arrival rate 2 µ2 = mean holding time Assume that the capacity of the link is inﬁnite: Is it a birth-death process? 41 / 47 Birth Processes Birth-Death Processes J. Virtamo Relationship to Markov Chains Linear Birth-Death Processes Examples 38.3143 Queueing Theory / Birth-death processes Call blocking in an ATM network Example 2. 11 Call blocking in an ATM network An ATM network(VP) of an ATMof two different types. diﬀerent types. A virtual path offers calls network is oﬀered calls of two R R 1 = 1Mbps 2 = 2Mbps R1 = 1Mbps R2 = 2Mbps λ1 = arrival rate λ2 = arrival rate µ2 = mean holding time µ1 = rate λ1 =arrivalmean holding time λ = arrival rate 2 µ1 a) The capacity of the link is large (inﬁnite) = mean holding time µ2 = mean holding time Assume that the capacity of the link is inﬁnite: n2 l2 l1 (n2+1)m2 l1 n1m1 l2 (n1+1)m1 n2m2 n1 The state variable of the Markov process in this example is the pair (N , N ), where N deﬁnes The state variable isi the pair (N1 , N2 ) where Ni deﬁnes 1 2 numberi of the the number of class- connections in progress. class-i connections in progress. 42 / 47 Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Call blocking in an ATM network An ATM network offers calls of two different types. R1 = 1Mbps λ = arrival rate 1 µ1 = mean holding time R2 = 2Mbps λ = arrival rate 2 µ2 = mean holding time Assume that the capacity of the link is limited to 4.5 Mbps Is it a birth-death process? 43 / 47 Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Call blocking in an ATM network An ATM network offers calls of two different types. J. Virtamo 38.3143 Queueing Theory / Birth-dea R1 = 1Mbps R2 = 2Mbps Call blocking in an ATM λ2 = arrival rate network (continued) λ = arrival rate 1 µ1 = mean holding time µ2 = mean holding time Assume that the capacity of the link is limited to 4.5 Mbps b) The capacity of the link is 4.5 Mbps n2 n1 44 / 47 Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Exercise 1 Process deﬁnition • There are two transatlantic cables each of which handle one telegraph message at a time. • The time-to-breakdown for each has the same exponential random distribution with parameter...
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