4.9. Consider the call center of Example 4.10. Suppose the arrival rate is 60 calls per...

 

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4.9. Consider the call center of Example 4.10. Suppose the arrival rate is 60 calls per hour and each call takes an average of 6 minutes to handle. If there are eight reservation agents and the system can put four callers on hold at one time, what is the probability that the system is full after 1 hour, assuming it is idle at time 0? Example 4.10. (Call Center). An airline phone-reservation system is called a call center and is staffed by s reservation clerks called agents. An incoming call for reservations is handled by an agent if one is available; otherwise the caller is put on hold. The system can put a maximum of H callers on hold. When an agent becomes available, the callers on hold are served in order of arrival. When all the agents are busy and there are H calls on hold, any additional callers get a busy signal and are permanently lost. Let X(t) be the number of calls in the system, those handled by the agents plus any on hold, at time t. Assume the calls arrive according to a PP(A) and the processing times of the calls are iid Exp(u) random variables. Model {X(t), t≥ 0} as a CTMC. The state space is {0, 1, 2,..., K}, where K = s + H. Suppose X(t) = i. For 0≤i≤K-1, the transition to state i + 1 is caused by an arrival, which occurs at rate A. In state K there are no arrivals. For 1 <i<K, the transition to state i - 1 is caused by the completion of the processing of any of the min(i,s) calls that are under service. The completion rate of an individual call is μ; hence the transition rate from state i to i - 1 is min(i,s)μ. Thus {X(t), t≥ 0} is a finite birth and death process with birth parameters λ=λ, 0≤i≤K-1, and death parameters Hi min(i,s), 0 ≤i≤K. The rate diagram is shown in Figure 4.8. We shall see in Chapter 6 that {X(t), t≥ 0} is called an M/M/s/K queue in queueing terminology. 0 M-1 M M+1 K-1 K μ 2μ (M-1)μ Μμ Μμ Μμ Μμ Μμ Fig. 4.8 Rate diagram of the call center. 4.9. Consider the call center of Example 4.10. Suppose the arrival rate is 60 calls per hour and each call takes an average of 6 minutes to handle. If there are eight reservation agents and the system can put four callers on hold at one time, what is the probability that the system is full after 1 hour, assuming it is idle at time 0? Example 4.10. (Call Center). An airline phone-reservation system is called a call center and is staffed by s reservation clerks called agents. An incoming call for reservations is handled by an agent if one is available; otherwise the caller is put on hold. The system can put a maximum of H callers on hold. When an agent becomes available, the callers on hold are served in order of arrival. When all the agents are busy and there are H calls on hold, any additional callers get a busy signal and are permanently lost. Let X(t) be the number of calls in the system, those handled by the agents plus any on hold, at time t. Assume the calls arrive according to a PP(A) and the processing times of the calls are iid Exp(u) random variables. Model {X(t), t≥ 0} as a CTMC. The state space is {0, 1, 2,..., K}, where K = s + H. Suppose X(t) = i. For 0≤i≤K-1, the transition to state i + 1 is caused by an arrival, which occurs at rate A. In state K there are no arrivals. For 1 <i<K, the transition to state i - 1 is caused by the completion of the processing of any of the min(i,s) calls that are under service. The completion rate of an individual call is μ; hence the transition rate from state i to i - 1 is min(i,s)μ. Thus {X(t), t≥ 0} is a finite birth and death process with birth parameters λ=λ, 0≤i≤K-1, and death parameters Hi min(i,s), 0 ≤i≤K. The rate diagram is shown in Figure 4.8. We shall see in Chapter 6 that {X(t), t≥ 0} is called an M/M/s/K queue in queueing terminology. 0 M-1 M M+1 K-1 K μ 2μ (M-1)μ Μμ Μμ Μμ Μμ Μμ Fig. 4.8 Rate diagram of the call center.

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