Waiting Line Approximation Spreadsheet Mean (average) Time Between Customer Arrivals = 1 Standard Deviation of the Time Between Customer Arrivals = 0.8485 Mean (average) Service Time= 5.4 Standard Deviation of the Service Time = 3 Number of Servers = 5 Customer Arrival Rate () = 1.00 Service Rate () = 0.19 Expected Server Utilization () = 1.0800 Coefficient of Variation of Customer Arrival Time = 0.8485 Coefficient of Variation of Service Time 0.5556 Expected Number of Customers Waiting in Line (Lq) = Infeasible Expected Time Waiting to be Served (Wq) = Infeasible mmk.xls M/M/k Queueing Formula Spreadsheet Inputs: lambda mu Definitions of terms: lambda = mu = k = Lq = L = Wq = W = P(0) = P(delay) = 15 5 arrival rate service rate number of servers average number in the queue average number in the system average wait in the queue average wait in the system probability of zero customers in the system ### probability that an arriving customer has to wait Outputs: k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Lq infinity infinity infinity 1.5283 0.3542 0.0991 0.0282 0.0078 0.0020 0.0005 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L infinity infinity infinity 4.5283 3.3542 3.0991 3.0282 3.0078 3.0020 3.0005 3.0001 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 Wq infinity infinity infinity 0.1019 0.0236 0.0066 0.0019 0.0005 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 W infinity infinity infinity 0.3019 0.2236 0.2066 0.2019 0.2005 0.2001 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 0.2000 P(0) 0.0000 0.0000 0.0000 0.0377 0.0466 0.0490 0.0496 0.0497 0.0498 0.0498 0.0498 0.0498 0.0498 0.0498 0.0498 0.0498 0.0498 0.0498 0.0498 0.0498 0.0498 0.0498 0.0498 0.0498 0.0498 P(delay) 1.0000 1.0000 1.0000 0.5094 0.2362 0.0991 0.0376 0.0129 0.0040 0.0012 0.0003 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 r ### 1.0000 1.0000 1.0000 0.7500 0.6000 0.5000 0.4286 0.3750 0.3333 0.3000 0.2727 0.2500 0.2308 0.2143 0.2000 0.1875 0.1765 0.1667 0.1579 0.1500 0.1429 0.1364 0.1304 0.1250 0.1200