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exactly one unit is placed each time the on-hand inventory decreases by one unit The lead times of the replenishments are independent and identically distributed random variables with mean Establish an equivalence with the Erlang loss model and give expressions for the long-run average on-hand inventory and the long-run fraction of demand that is lost 514 In an electronic system there are c elements of a crucial component connected in parallel to increase the reliability of the system Each component is switched on and the lifetimes of the components have an exponential distribution with mean 1/ The lifetimes of the components are independent of each other The electronic system is working as long as at least one of the components is functioning, otherwise the system is down A component that fails is replaced by a new one It takes an exponentially distributed time with mean 1/ to replace a failed component Only one failed component can be replaced at a time (a) Use a continuous-time Markov chain to calculate the long-run fraction of time the system is down Specify the transition rate diagram rst (b) Does the answer in (a) change when the replacement time of a failed component has a general probability distribution with mean 1/ (Hint : compare the transition rate diagram with the transition rate diagram in the Erlang loss model) 515 Reconsider Exercise 514 but this time assume there are ample repairmen to replace failed components (a) Use a continuous-time Markov chain to calculate the long-run fraction of time the system is down Specify the transition rate diagram rst (b) What happens to the answer in (a) when the replacement time is xed rather than exponentially distributed (Hint : compare the transition rate diagram with the transition rate diagram in the Engset loss model) 516 Suppose you have two groups of servers each without waiting room The rst group consists of c1 identical servers each having an exponential service rate 1 and the second group consists of c2 identical servers each having an exponential service rate 2 Customers for group i arrive according to a Poisson process with rate i (i = 1, 2) A customer who nds all servers in his group busy upon arrival is served by a server in the other group, provided one is free, otherwise the customer is lost Show how to calculate the long-run fraction of customers lost 517 Consider a conveyor system at which items for processing arrive according to a Poisson process with rate The service requirements of the items are independent random variables having a common exponential distribution with mean 1/ The conveyor system has two work stations 1 and 2 that are placed according to this order along the conveyor Workstation i consists of si identical service channels, each having a constant processing rate of i (i = 1, 2); that is, an item processed at workstation i has an average processing time of 1/( i ) Both workstations have no storage capacity and each service channel can handle only one item at a time An arriving item is processed by the rst workstation in which a service channel is free and is lost when no service channel is available at either of the stations Show how to calculate the fraction of items lost and solve for the numerical data = 10, = 1, 1 = 2, 2 = 15, s1 = 5 and s2 = 5 (Answer: 00306) Verify experimentally that the loss probability is nearly insensitive to the distributional form of the service requirement (eg compute the loss probability 00316 for the data when the service requirement has an H2 distribution with balanced means and a squared coef cient of variation of 4) 518 Consider a stochastic service system with oisson arrivals at rate and two different groups of servers, where each arriving customer simultaneously requires a server from both groups An arrival not nding that both groups have a free server is lost and has no further in uence on the system The ith group consists of si identical servers (i = 1, 2) and each server can handle only one customer at a time An entering customer occupies the two assigned servers from the groups 1 and 2 during independently exponentially distributed times with respective means 1/ 1 and 1/ 2 Show how to calculate the loss probability.

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To illustrate this, consider the single-server M X /G/1/N + 1 queue with partial rejection For the M X /G/1 queue the asymptotic expansion j( ) j as j holds when the service time is not heavy-tailed, where the constants and = 1/ are determined by the relations (935) and (936) When using the asymptotic expansion one needs only to calculate the root of a non-linear equation in a single variable Two-moment approximation The practical applicability of the formulas for Prej stands or falls with the computation of the state probabilities j( ) In some queueing models it is computationally feasible to calculate these probabilities sing embedded Markov chain analysis or continuous-time Markov chain analysis However, in many queueing models the exact computation of the state probabilities j( ) is not practically feasible This.

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where Ndet ( ) and Nexp ( ) are the (approximate) values of the minimal buffer size N ( ) for the M X /D/c/c + N queue and the M X /M/c/c + N queue The buffer sizes Ndet ( ) and Nexp ( ) are computed by using the (approximate) formula for Prej in the particular cases of deterministic services and exponential services Relatively simple algorithms are available to compute the state probabilities j( ) in the M X /M/c queue and the M X /D/c queue; see Section 963 The two-moment 2 approximation (9814) is only recommended when cS is not too large (say, 0 2 cS 2) Table 984 illustrates the performance of the two-moment approximation (9814) for the M/G/c/c+N queue, where the number of servers has the two values c = 1 2 and c = 10 For both Erlang-2 services (cS = 05) and H2 services with gamma 2 = 2), the approximate and exact values of N ( ) are given for normalization (cS several values of Any fractional value resulting from the interpolation formula (9814) has been rounded up The results in the table show an excellent performance of the two-moment approximation and also nicely demonstrate that N ( ) increases logarithmically in as increases 984 Discrete-Time Queueing Systems Many practical queueing systems operate on a discrete-time basis A discrete-time queueing system is characterized by time-slotted service A new service can only start at the beginning of a time slot, and the service time is a multiple of time slots In applications the discrete-time queueing systems typically have nite buffers to store incoming packets Packets are the entities to be served Let us assume that there are c service channels and a buffer of capacity N to store incoming packets The buffer excludes any packet in service Each service channel can handle only one packet at a time A new service can only start at the beginning of a time slot The service times of he packets are independent of each other It is.

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