3 Question 1 15 pts Consider 20 DS0 circuits that are multiplexed onto 3 outgoing circuits. The mean arrival rate at each DS0 circuit is 1 call per hour during the busy hour; calls last, on average, 5 minutes. Assume Poisson arrivals, exponentially distributed call durations, and that customers that attempt a call when all circuits are busy are blocked (ignore call retry). The probability of blocked calls is. (Use 2 digits to the right of the decimal point.) - Poisson arrivals at a rate of 20 calls/hour (1/hour on each of 20 incoming circuits). That's Lambda. - Exponentially distributed service times at a rate of 12 calls/hour for each server (1 call every 5 min == 12 calls/hour). That's Mu. - 3 outgoing circuits == 3 servers - No queue ("customers that attempt a call when all circuits are busy are blocked "), so the maximum number of clients in the system == 3 Based on the above, the queuing model is M/M/3/3. Now, take a look at lecture IIB3_OtherMMResults, slides 8 - 10. This is a straightforward Erlang B system (M/M/m/m, where m==3). The question asked is "what is the probability of blocked calls?", so what is the probability that all servers are busy [see slide 10]? So the process is: 1. Determine the appropriate queuing model 2. Determine Lambda, Mu, and other variables 3. Determine what the question is asking 4. Apply the appropriate formula Question 2 5 pts Customers arrive at a two-chair shoeshine stand at a rate of 10 customers per hour. The average length of a shoeshine is 5 minutes. There is only one attendant, so that one chair is used as a waiting position. Customers who find both chairs occupied go away. Assume Poisson arrivals and exponential service times. The queuing model that is appropriate here is . Question 3 5 pts In question 2, the probability that the waiting chair will be occupied is . (Use two digits to the right of the decimal point.) Question 4 10 pts In question 2, the mean number of customers served per hour is . (Use two digits to the right of the decimal point.) Question 5 10 pts Now, let us modify question 2 by assuming there are 2 attendants at the stand. The mean number of customers serviced per hour is . (Use two digits to the right of the decimal point.) Question 6 10 pts An entrepreneur offers services that can be modeled as an s-server Erlang loss (Erlang-B) system. Suppose the arrival rate is 4 customers per hour; the average service time is 1 hour; the entrepreneur earns $ 2.50 for each customer served; and the entrepreneur's operating cost is $1.00 per server per hour (whether the server is busy or idle). The optimal number of servers, from the entrepreneur's point of view is . The way to approach problem 6-8 (at least my simple approach) was to make a table as follows: _Num Servers_ ____Blocking Probability____ ___Profit___ 3 (calculate using Erlang-B) $X.XX 4 5 6, etc With enough rows you can answer all three of those questions. (If you have more than 10 rows, you are doing something wrong.) The 1-hour "profit" formula is simply the # Customers served per hour * $2.50, minus $1 per server. Again, if you know the blocking probability, you can easily calculate the mean number of customers served per hour. Remember, mean # customers blocked (or 'not served') per hour is the blocking probability multiplied by the mean arrival rate. The rest of them are served. I should also mention that the number of customers served is an average (mean), so it is NOT going to be a whole number. Edited by David Richard Raymond on Feb 5 at 11:31am Question 7 5 pts In question 6, if the entrepreneur deploys the optimal number of servers, he/she should expect to make a profit of $ per hour. (Use two digits to the right of the decimal point.) Question 8 10 pts In question 6, the maximum number of servers beyond which it is unprofitable for the entrepreneur to remain in business is . Question 9 0 pts You may upload your work here for questions 1 through 8. I'll be able to review and give partial credit if required. If you have trouble with the upload, email files to me at raymondd@vt.edu. Upload