4Com sells routers, hubs and switches to both businesses and consumers. They ope

Question

4Com sells routers, hubs and switches to both businesses and consumers. They operate a technical support telephone center for their business customers. Here are the steps when a customer calls:

i)Every customer who calls may be placed in a line. The line has no limit, and customers are infinitely patient.

ii) After the (possible) line, the customer first speaks with a Service Agent (or ‘Agent’). The agent gathers basic information, but occasionally, the agent can solve the problem for the customer. In that case, the customer leaves the system. If not, the customer is passed along to a Specialist. About 60% of customers are passed to a Specialist (40% leave the system after talking with the agent). Service times do not seem to depend on whether a customer call ends with the agent; averaging over all customers, customers spend 20 minutes with an Agent, with a standard deviation of 20 minutes.

iii) If a customer reaches a Specialist, the Specialist makes sure that the customer is satisfied before ending the call. On average, a customer who reaches a Specialist spends 30 minutes talking with the Specialist, with a standard deviation of 15 minutes.

Last month during the peak times, calls arrived at a rate of 12 per hour, with an arrival CV of 1. They employed 5 Agents and 4 Specialists, but customers were not happy with the level of service. This month 4Com expects a similar arrival rate.

a) First consider those customers who only see Agents (they do not go on to Specialists). On average, how long will a customer wait for an Agent?

b) Now consider customers who see both Agents and Specialists. On average, how long does this customer wait in line?

c) Find the average time that an arriving customer is in the system (averaged over all customers).

Explanation / Answer

a) First consider those customers who only see Agents (they do not go on to Specialists). On average, how long will a customer wait for an Agent?

About 60% customers pass on to Specialists so arrival rate for specialists is 7.2 per hour and specialist takes 30 minutes so service rate is 2 per hour. There are 4 specialists, the queue is analized as follows:

Average time in the queue 59 minutes and average time in the system approx. one and a half hours seems to be too high that is the complaint from the customers.

Unable to write down the complex / complicated formulas, please refer to book.

Parameter Value Parameter Value Minutes Seconds M/M/s Average server utilization 0.8 Arrival rate(lambda) 12 Average number in the queue(Lq) 2.2165 Service rate(mu) 3 Average number in the system(Ls) 6.2165 Number of servers 5 Average time in the queue(Wq) 0.1847 11.0823 664.9351 Average time in the system(Ws) 0.518 31.0823 1,864.94

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