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521 Consider the following modi cation of Example 542 Instead of in nite-source input, there is nite-source input for each of the two message types The source of messages of type j has Mj users, where each user generates a new message after an exponentially distributed think time with mean 1/ j provided the user has no message in service at the communication system Assume the numerical data c = 10, M1 = M2 = 10, 1 = 3, 2 = 1, 1 = 4, 2 = 1 Use continuous-time Markov chain analysis to compute the L-policy for which the average throughput is maximal Does the result change when the transmission times are constant rather than exponentially distributed 522 Suppose a production facility has M operating machines and a buffer of B standby machines Machines in operation are subject to breakdowns The running times of the operating machines are independent of each other and have a common exponential distribution with mean 1/ An operating machine that breaks down is replaced by a standby machine if one is available A failed machine immediately enters repair There are ample repair facilities so that any number of machines can be repaired simultaneously The repair time of a failed machine is assumed to have an exponential distribution with mean 1/ For given values of , and M, demonstrate how to calculate the minimum buffer size B in order to achieve that the long-run fraction of time that less than M machines are operating is no more than a speci c value Do you expect the answer to depend on the speci c form of the repair-time distribution 523 Suppose a communication system has c transmission channels at which messages arrive according to a Poisson process with rate Each message that nds all of the c channels busy is lost upon arrival, otherwise the message is randomly assigned to one of the free channels The transmission length of an accepted message has an exponential distribution with mean 1/ However, each separate channel is subject to a randomly changing environment that in uences the transmission rate of the channel Independently of each other, the channels alternate between periods of good condition and periods of bad condition These alternating periods are independent of each other and have exponential distributions with means 1/ g and 1/ b The transmission rate of a channel being in good (bad) condition is g ( b ) Set up the balance equations for calculating the fraction of messages that are lost Noting that = ( b g + g b )/( g + b ) is the average transmission rate used by a channel, make some numerical comparisons with the case of a xed transmission rate 524 Jobs have to undergo tooling at two stations, 1 and 2, which are linked in series New jobs arrive at station 1 according to a Poisson process with rate At station 1 they undergo their rst tooling Upon completion of the tooling at station 2, there is a given probability p that both toolings have to be done anew In this case the job rejoins the queue at station 1, otherwise the job leaves the system The handling times of a job at stations 1 and 2 are independent random variables having exponential distributions with respective means 1/ 1 and 1/ 2 Each station can handle only one job at a time What is the long-run average amount of time spent in the system by a newly arriving job 525 Consider a closed queueing network as in Section 562 Assume now that the service rate at station i is a function i (ni ) of the number (ni ) of customers present at station i Verify that the product-form solution is given by.

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526 Consider the M/G/1 queue with Erlangian services from Example 551 De ne the generating functions (z) = j =1 j zj and F (z) = j =0 fj zj Let R be the convergence radius of the series j =1 j zj It is assumed that R > 1

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Table 62 CAMEL phase 4 subsets Description The VMSC or GMSC is capable of initiating CAMEL phase 4 services for MO and MF calls The VMSC or GMSC is capable of initiating CAMEL phase 4 subscribed dialled services The VMSC is capable of initiating CAMEL phase 4 services for MT calls The VMSC or SGSN has the capability to trigger a CAMEL phase 4 service for MT SMS The GMSC is capable of initiating CAMEL phase 4 services for MT calls The SGSN support GPRS mobility management The SGSN is capable of receiving MAP provide subscriber info (PSI) PSI is used for the ATI procedure

F (z) =

f0 (1 z) (1 z) z(1 (z))

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(b) Use Theorem C1 in Appendix C to prove that fj j as j for some constant , where is the reciprocal of the smallest root of (1 x) x(1 (x)) = 0 on (1, R) (c) Verify that 1 Wq (x) 1 e (1 )x as x for some constant 1 > 0 527 Consider the so-called MAP /G/1 queue with a Markov modulated Poisson arrival process (an important application of this model in teletraf c analysis is the buffering of independent on-off sources at a statistical multiplexer) The arrival rate of customers is governed by an exogenous phase process The phase process is a continuous-time Markov chain with nitely many states s = 0, 1, , m and in nitesimal transition rates st It is assumed that the phase process is irreducible and thus has a unique equilibrium distribution which is denoted by {es } If the phase process is in state s, customers arrive according to a Poisson process with rate s The service times of the customers are independent random variables which are also independent of the arrival process Customers are served in order of arrival It is assumed that the service time of a customer has the same probability distribution m e 1 j , it is function (553) as in Example 551 Letting = j =1 j s=0 s s assumed that the server utilization is less than 1 Also it is assumed that the convergence radius R of the power series (z) = j =1 j zj is larger than 1 (a) Let p(i, s) denote the joint equilibrium probability that i customers are present and the arrival process is in phase s Verify that for any s there is a constant s such that p(i, s) s i as i ,

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