formula sheets

R= (Avg. Number of customers being served) P= (System Utilization)
Lq= (Avg. number of customers in line) Single server
Ls=Lq+R (Average number of customers in system)
Ws=Wq+ (Average time in the system) Ws= single server Wq=
Pn=(1-P) (Probability of exactly N customers in system) single server (Probability of at least N customers in system) single server
P (Probability of waiting in queue longer than t) single server (Probability of waiting in system longer than t) single server
Lq (lookup in table for multiple servers)
Pw=( (Probability a new arrival will have to wait)
Pw (Probability of waiting in queue longer than t) multiple server
Notations: λ=Average arrival rate μ=Average service rate Prob of zero wait=(1-p)
Examples:
1. Mechanics arrive at an average rate of 40/hr. Clerk can fill requests in 3 minutes on average (60/3=20/hr). Clerks are paid $6 per hr and mechanics are paid $12 per hr. What is optimal number of clerks to staff the counter? λ=40/hr μ=20/hr =2 Waiting cost + service cost=Total cost
M=3
Service cost = 3*6=18 R
Waiting cost= $12* Ls
Lq=.889 (table) Ls=.889+2=2.889 Waiting cost=$12*2.889=34.668

2. “If you wait more than 5 mins in queue, we will give you 1000 flyer miles”. An avg. of 120 customers arrive per hr and each takes on avg. 1 min to get through. If each mile is valued at 2 cents, and the hourly staff cost for each gate is $30. How many gates should be open? λ =120/hr μ=60/hr =2
M=3 Service cost= $30*3=90 Lq=.889 (table) Wq=.889/120 Pw=(3(60)-120)(.889/120)=.4445
Prob(waiting in queue ≥5min)=.4445*=.00299
1000*(2/1000)*120*.00299=7.19 (Waiting cost) TC=7.19+90=97.19