Case 4 Logistic challenges

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Single-Server

Single-Server Model
Arrival Rate (λ) 25
Service Rate (μ) 30
Probability of zero customers in the system (P0) 0.1666666667
Probability of exactly 10 customers in the system 0.0269175971
Probability of <= 10 customers in the system 0.8654120143
Probability of > 10 customers in the system 0.1345879857
Average utilization of the server (ρ) 0.8333333333
Average number of customers in the system (L) 5
Average number of customers in the line (Lq) 4.1666666667
Average time in the system (W) 0.2
Average waiting time in the line (Wq) 0.1666666667
Average time being served 0.0333333333

The data showing when you open this spreadsheet is from the Phlebotomist example in the PowerPoint: λ = 25 Pts per hour  = 30 Pts per hour

Multi-Server

Multi-Server Model (with S <= 4)
Number of Servers (S) 2
Arrival Rate (λ) 50
Service Rate (μ) 30
Probability of zero customers in the system (P0) 0.091 PROBABILITY TABLE
Probability of exactly 10 customers in the system 0.029 n Pr(x=n) Pr(x<=n) Pr(x>n)
Probability of <= 10 customers in the system See Table 0 0.091 0.091 0.909
Probability of > 10 customers in the system See Table 1 0.152 0.242 0.758
Average utilization of the server (ρ) 0.833 2 0.126 0.369 0.631
Average number of customers in the system (L) 5.455 3 0.105 0.474 0.526
Average number of customers in the line (Lq) 3.788 4 0.088 0.562 0.438
Average time in the system (W) 0.109 5 0.073 0.635 0.365
Average waiting time in the line (Wq) 0.076 6 0.061 0.696 0.304
Average time being served 0.033 7 0.051 0.746 0.254
8 0.042 0.789 0.211
9 0.035 0.824 0.176
10 0.029 0.853 0.147
11 0.024 0.878 0.122
12 0.020 0.898 0.102
13 0.017 0.915 0.085
14 0.014 0.929 0.071
15 0.012 0.941 0.059
16 0.010 0.951 0.049
17 0.008 0.959 0.041
18 0.007 0.966 0.034
19 0.006 0.972 0.028
20 0.005 0.976 0.024
21 0.004 0.980 0.020
22 0.003 0.984 0.016
23 0.003 0.986 0.014
24 0.002 0.989 0.011
25 0.002 0.990 0.010

The data showing when you open this spreadsheet is from the Phlebotomist example in the PowerPoint: 2 Phlebotomists (servers) λ = 50 Pts per hour  = 30 Pts per hour

Sheet3