The Outfront BBQ Rib Haven does carry out only. During peak periods, two servers

Question

The Outfront BBQ Rib Haven does carry out only. During peak periods, two servers are on duty. The owner notices that during these periods, the percentage of idle time of each server is 1%. Assume that probabilities of joining the servers are all equal. If the owner decides to add a third server during these times, how much idle time would each server have then? Suppose that by adding the third server, the pressure on the servers is reduced, so they can work more carefully, but their service output rate is reduced by 20%. What now is the percent time each would be idle? In the following, consider a G/G/c queue with identical servers. Show that r is equal to the expected number of customers being served. Show that p is equal to the busy probability of a server.

Explanation / Answer

a) probaility that the server is idle (=1-lamda/mu) 0.01 when 2 servers are on duty with three servers, new lamda= (3/2) lamda, and new mu= 3/2(mu)) hence for 3 servers, probaility that the server is idle = 1 - ((3/2)lamda/(3/2) mu) probaility that the server is idle =1-lamda/mu =0.01 hence, the idle time wont change b) service rate = 0.8 * mu (20% reduction in service rate) then the idle time = 1 - ((3/2)lamda/(0.8 mu) 1-(lamda/mu) = 0.01 Lamda/mu = 1-0.01 = 0.99 r = (3/2)lamda/(0.8)mu = (3/2)*(1/0.8)*(0.99) = 1.85625 (r=lambda / mu) & rho = r/c (c=number of counters)) ………………..(question2) rho = r/c = 0.61875 (c=3) server is busy 61.87 % time, hence idle time = 1-0.62 i.e. 38% hence, 38% time the server will be idle

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