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(Hint : x j and assume that the process incurs a cost at rate 1 whenever j customers are present and a cost at rate 0 otherwise Imagine that the server continues servicing ctitious customers when the system is empty so that actual or ctitious service completions occur according to a Poisson process with rate ) 319 In each time unit a job arrives at a conveyor with a single workstation The workstation can process only one job at a time and has a buffer with ample capacity to store the arriving jobs that nd the workstation busy The processing times of the jobs are independent random variables having a common Erlang (r, ) distribution It is assumed that r/ < 1 (a) De ne an appropriate Markov chain to analyse the number of jobs in the buffer just prior to the arrival epochs of new jobs and specify the one-step transition probabilities (b) Explain how to calculate the long-run average delay in the buffer per job (c) Prove that the equilibrium distribution of this Markov chain has a geometric tail 320 Consider Exercise 319 again but now assume that the buffer has nite capacity Any arriving job that nds the buffer full is lost Show how to calculate the long-run fraction of lost jobs and the long-run fraction of time the workstation is busy (Hint : use Little s formula for the latter performance measure) 321 At the telephone exchange, calls arrive according to a Poisson process with rate The calls are rst put in an in nite-capacity buffer before they can be processed further The buffer is periodically scanned every T time units, and only at those scanning epochs are calls in the buffer allocated to free transmission lines There are c transmission lines and each transmission line can handle only one call at a time The transmission times of the calls are independent random variables having a common exponential distribution with mean 1/ (a) Use Markov chain analysis to nd the equilibrium distribution { j } of the number of calls in the buffer just prior to the scanning epochs (b) Argue that the long-run average number of calls in the buffer is given by.

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(Hint : imagine that each call is marked upon arrival and is unmarked at the next scanning epoch Argue that the average number of marked calls in the buffer is 1 T ) 2 (c) What is the long-run average delay in the buffer per call 322 Consider Example 341 with Poisson arrivals of messages c 1 (a) Prove the validity of the relation = j =1 j j + c j =c j and note that this relation can be used as an accuracy check on the calculated values of the state probabilities j , j = 0, 1, (b) Use the hint in Exercise 321 to prove that the long-run average number of messages in the buffer equals j =c+1 (j c) j + 1 T 2 (c) What is the long-run average delay in the buffer per message 323 Consider Example 341 again but assume now that the buffer for temporarily storing arriving messages has a nite capacity K Each arriving message that nds the buffer full is lost

(a) Modify the one-step transition probabilities of the Markov chain {Xn } describing the number of messages in the buffer at the end of the time slots (K) (b) Denoting by { j , j = 0, 1, , K} the equilibrium distribution of the Markov chain, argue that the long-run fraction of messages lost is c 1 K 1 (K) (K) loss (K) = j j c j

(Hint : the sum of the average number of messages lost per time unit and the average number of messages transmitted per time unit equals ) (c) Let K( ) be the smallest value of K for which loss (K) for a given value of Letting = /c, compute for = 090, 095 and c = 1, 5, 10 the values of K( ) as given in the table below Note that K( ) increases logarithmically in as increases What does this mean for the asymptotic behaviour of loss (K) as K gets large = 080 10 6 10 8 10 10 c=1 29 40 50 c=5 32 42 53 c = 10 36 46 57 c=1 107 152 197 = 095 c=5 110 155 200 c = 10 114 159 204

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