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Vn (N ) = Cf + Vn 1 (N + 1), Vn (N + 1) = 0 + Vn 1 (1) We have applied the standard value-iteration algorithm to the numerical data from Table 641 For each stationary policy the associated Markov chain {Xn } is aperiodic Taking V0 (i) = 0 for all i and the accuracy number = 10 3 , the algorithm is stopped after n = 28 iterations with the stationary policy R(n) = (0, 0, 0, 1, 2, 2) together with the lower and upper lower bounds mn = 04336 and Mn = 04340 The average cost of policy R(n) is estimated by 1 (mn + Mn ) = 2 04338 and this cost cannot deviate more than 01% from the theoretically minimal average cost In fact policy R(n) is optimal as we know from previous results obtained by policy iteration To get a feeling of how strongly the required number of iterations depends on , we applied standard value-iteration for = 10 2 and = 10 4 as well For these choices of the accuracy number , standard valueiteration required 21 and 35 iterations respectively Example 661 A nite-capacity queue with deterministic arrivals Consider a single-server queueing system having a nite waiting room for K customers (including any customer in service) The arrival rocess of customers is deterministic Every D time units a customer arrives A customer nding a full waiting room upon arrival is lost The service times of the customers are independent random variables having an Erlang (r, ) distribution What is the long-run fraction of customers who are lost Taking the constant interarrival time as time unit, the fraction of lost customers can be seen as an average cost per time unit when a cost of 1 is incurred each time an arriving customer nds the waiting room full The queueing process embedded at the arrival epochs can be described by a Markov process by noting that the Erlang (r, ) distributed service time can be seen as the sum of r independent phases each having an exponential distribution with mean 1/ A customer is served by serving its phases one at a time The queueing problem can now be converted into a Markov decision model with a single action in each state The state of the system is observed at the arrival epochs and the set of possible states of the system is given by I = {0, 1, , Kr} State i corresponds to the situation that i uncompleted service phases are present just prior to the arrival of a new customer In each state i there is a single action to be denoted by a = 0 The action a = 0 in state i corresponds to the acceptance of the newly arriving customer when i Kr r and corresponds to the rejection of the customer otherwise The one-step costs ci (a) are given by ci (a) = 0 1 if i Kr r, if i > Kr r.

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The kth packet pik of ow i is assigned a starttime tag S(pik ) and a nish- time tag F(pik ) according to the following algorithm: div>. The lag of a ow is computed as the difference between the error-free service and real service received by he ow. In this case, a ow that falls. 128 data on word documents to encode barcode 128 and .Related: Code 39 Generation .NET WinForms , Print ITF-14 ASP.NET , Create Codabar C#

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and b = 20 Try other numerical examples and investigate whether the optimal control rule is characterized by integers L0 , , LM so that an arriving unit of raw material nding i units present at station 1 is only accepted when less than Li units are present at station 2 78 Consider the situation that two groups of servers share a common waiting room The rst group consists of c1 servers and the second group consists of c2 servers Customers for the rst group arrive according to a Poisson process with rate 1 and, independently of this process, customers for the second group arrive according to a Poisson process with rate 2 Upon arrival of a new customer, a decision has to be made to accept or reject them An accepted customer keeps one place in the waiting room occupied until their service is completed The service times of he customers are exponentially distributed with a mean 1/ 1 for a customer going to the rst group and mean 1/ 2 for a customer going to the second group Each server can handle only one customer at a time and serves only customers for the group to which the server belongs The goal is to nd a control rule minimizing the total average rejection rate Develop a value-iteration algorithm for this control problem Solve for the numerical data c1 = c2 = 1, 1 = 12, 2 = 1, 1 = 2 = 1 and M = 15, where M denotes the number of places in the waiting room Try other numerical examples and verify experimentally that the optimal control rule is characterized by two sequences (r) (r) {a1 , 0 r M} and {a2 , 0 r M} so that an arriving customer of type k nding r customers of the other type present upon arrival is accepted only if less than ar customers of the same type k are present and the waiting room is not full This problem is based on Tijms and Eikeboom (1986) 79 Consider the problem of designing an optimal buffer management policy in a sharedmemory switch with the feature that packets already accepted in the switch can be dropped (pushed out) The system has two output ports and a nite buffer shared by the two output ports Packets of types 1 and 2 arrive according to independent Poisson processes with rates 1 and 2 Packets of type i are destined for output port i for i = 1, 2 At each of the two output ports there is a single transmission channel Each channel can transmit only one packet at a time and the transmission time at output port i is exponentially distributed with mean 1/ i for i = 1, 2 Upon arrival of a new packet, the system has to decide whether to accept the packet, to reject it, or to accept it and drop a packet of the other type A packet that is rejected or dropped is called a lost packet and has no further in uence on the system The total buffer size is B and it is assumed that an accepted packet occupies a buffer place until its transmission is completed The goal is to nd a control rule minimizing the overall fraction of packets that are lost Develop a value-iteration algorithm Solve for the numerical data 1 = 1, 2 = 10, 1 = 2, 2 = 20 and B = 12 Try other numerical examples and investigate whether the optimal control rule has a speci c structure This problem is based on Cidon et al (1995) 710 Consider a two-server facility with heterogeneous servers The faster server is always available and the slower server is activated for assistance when too many customers are waiting Customers arrive according to a Poisson process with rate The service facility has ample waiting room The service times of the customers are independent of each other and have an exponential distribution The mean service time is 1/ 1 when service is provided by the faster server and is 1/ 2 for the slower server, where 1/ 1 < 1/ 2 It is assumed that the load factor /( 1 + 2 ) is less than 1 The slower server can only be turned off when it has completed a service The slower server cannot be on when the system is empty If the slower server is kept on while customers are waiting for service, it cannot remain idle Each server can handle only one customer at a time Service is non-pre-emptive; that is, the faster server cannot take over a customer from the slower server A xed cost of K 0 is incurred each time the slower server is turned on and there is an operating cost of r > 0 per time unit the slower server is on Also, a holding cost of h > 0 per time unit is incurred for each customer in the system The goal is to nd a switching rule that minimizes the.

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