QUEUING ANALYSIS/LOGISTICS CHALLENGES
bandrade1981
MOD4CASE4.xlsxSingle-Server
| Single-Server Model | |||||||||||||
| 6 Trucks in System | |||||||||||||
| Arrival Rate (λ) | 3.5 | Arrival Rate (λ) | 3.5 | ||||||||||
| Service Rate (μ) | 4.2 | Service Rate (μ) | 4.2 | ||||||||||
| 5 Trucks In System | |||||||||||||
| Probability of zero customers in the system (P0) | 0.1666666667 | Probability of zero customers in the system (P0) | 0.1666666667 | ||||||||||
| Probability of exactly | 5 | Trucks in the system | 0.0669795953 | Probability of exactly | 6 | Trucks in the system | 0.0558163294 | ||||||
| Probability of <= | 5 | Trucks in the system | 0.6651020233 | Probability of <= | 6 | Trucks in the system | 0.7209183528 | ||||||
| Probability of > | 5 | Trucks in the system | 0.3348979767 | Probability of > | 6 | Trucks in the system | 0.2790816472 | ||||||
| Average utilization of the server (ρ) | 0.8333333333 | Average utilization of the server (ρ) | 0.8333333333 | ||||||||||
| Average number of customers in the system (L) | 5 | Average number of customers in the system (L) | 5 | ||||||||||
| Average number of customers in the line (Lq) | 4.1666666667 | Average number of customers in the line (Lq) | 4.1666666667 | ||||||||||
| Average time in the system (W) | 1.4285714286 | Average time in the system (W) | 1.4285714286 | ||||||||||
| Average waiting time in the line (Wq) | 1.1904761905 | Average waiting time in the line (Wq) | 1.1904761905 | ||||||||||
| Average time being served | 0.2380952381 | Average time being served | 0.2380952381 | ||||||||||
| 7 Trucks in System | Higherst Number of Trucks 95% Probability Rate | ||||||||||||
| Arrival Rate (λ) | 3.5 | ||||||||||||
| Service Rate (μ) | 4.2 | Arrival Rate (λ) | 3.5 | ||||||||||
| Service Rate (μ) | 4.2 | ||||||||||||
| Probability of zero customers in the system (P0) | 0.1666666667 | ||||||||||||
| Probability of exactly | 7 | Trucks in the system | 0.0465136079 | Probability of zero customers in the system (P0) | 0.1666666667 | ||||||||
| Probability of <= | 7 | Trucks in the system | 0.7674319606 | Probability of exactly | 16 | Trucks in the system | 0.0090146488 | ||||||
| Probability of > | 7 | Trucks in the system | 0.2325680394 | Probability of <= | 16 | Trucks in the system | 0.9549267559 | ||||||
| Average utilization of the server (ρ) | 0.8333333333 | Probability of > | 16 | Trucks in the system | 0.0450732441 | ||||||||
| Average number of customers in the system (L) | 5 | Average utilization of the server (ρ) | 0.8333333333 | ||||||||||
| Average number of customers in the line (Lq) | 4.1666666667 | Average number of customers in the system (L) | 5 | ||||||||||
| Average time in the system (W) | 1.4285714286 | Average number of customers in the line (Lq) | 4.1666666667 | ||||||||||
| Average waiting time in the line (Wq) | 1.1904761905 | Average time in the system (W) | 1.4285714286 | ||||||||||
| Average time being served | 0.2380952381 | Average waiting time in the line (Wq) | 1.1904761905 | ||||||||||
| Average time being served | 0.2380952381 | ||||||||||||
| New Service Rate at 95% Probability with 5 trucks or less | |||||||||||||
| Arrival Rate (λ) | 3.5 | ||||||||||||
| Service Rate (μ) | 5.9 | ||||||||||||
| Probability of zero customers in the system (P0) | 0.406779661 | ||||||||||||
| Probability of exactly | 5 | Trucks in the system | 0.0298840458 | ||||||||||
| Probability of <= | 5 | Trucks in the system | 0.9564190998 | ||||||||||
| Probability of > | 5 | Trucks in the system | 0.0435809002 | ||||||||||
| Average utilization of the server (ρ) | 0.593220339 | ||||||||||||
| Average number of customers in the system (L) | 5 | ||||||||||||
| Average number of customers in the line (Lq) | 2.9661016949 | ||||||||||||
| Average time in the system (W) | 0.4166666667 | ||||||||||||
| Average waiting time in the line (Wq) | 0.2471751412 | ||||||||||||
| Average time being served | 0.1694915254 | ||||||||||||
λ =3.5 Arrival Rate of trucks for hour = 4.2 Service Rates per truck hour
Multi-Server
| Multi-Server Model | (with S <= 4) | |||||||||
| Number of Servers (S) | 2 | |||||||||
| Arrival Rate (λ) | 3.5 | |||||||||
| Service Rate (μ) | 7 | |||||||||
| Probability of zero customers in the system (P0) | 0.600 | PROBABILITY TABLE | ||||||||
| Probability of exactly | 5 | Trucks in the system | 0.001 | n | Pr(x=n) | Pr(x<=n) | Pr(x>n) | |||
| Probability of <= | 5 | Trucks in the system | See Table | 0 | 0.600 | 0.600 | 0.400 | |||
| Probability of > | 5 | Trucks in the system | See Table | 1 | 0.300 | 0.900 | 0.100 | |||
| Average utilization of the server (ρ) | 0.250 | 2 | 0.075 | 0.975 | 0.025 | |||||
| Average number of customers in the system (L) | 0.533 | 3 | 0.019 | 0.994 | 0.006 | |||||
| Average number of customers in the line (Lq) | 0.033 | 4 | 0.005 | 0.998 | 0.002 | |||||
| Average time in the system (W) | 0.152 | 5 | 0.001 | 1.000 | 0.000 | |||||
| Average waiting time in the line (Wq) | 0.010 | 6 | 0.000 | 1.000 | 0.000 | |||||
| Average time being served | 0.143 | 7 | 0.000 | 1.000 | 0.000 | |||||
| 8 | 0.000 | 1.000 | 0.000 | |||||||
| 9 | 0.000 | 1.000 | 0.000 | |||||||
| 10 | 0.000 | 1.000 | 0.000 | |||||||
| 11 | 0.000 | 1.000 | 0.000 | |||||||
| 12 | 0.000 | 1.000 | 0.000 | |||||||
| 13 | 0.000 | 1.000 | 0.000 | |||||||
| 14 | 0.000 | 1.000 | 0.000 | |||||||
| 15 | 0.000 | 1.000 | 0.000 | |||||||
| 16 | 0.000 | 1.000 | 0.000 | |||||||
| 17 | 0.000 | 1.000 | 0.000 | |||||||
| 18 | 0.000 | 1.000 | 0.000 | |||||||
| 19 | 0.000 | 1.000 | 0.000 | |||||||
| 20 | 0.000 | 1.000 | 0.000 | |||||||
| 21 | 0.000 | 1.000 | 0.000 | |||||||
| 22 | 0.000 | 1.000 | 0.000 | |||||||
| 23 | 0.000 | 1.000 | 0.000 | |||||||
| 24 | 0.000 | 1.000 | 0.000 | |||||||
| 25 | 0.000 | 1.000 | 0.000 |
2 Loading Teams λ = 3.5 Arrival rate per hour = 7 Service rate per hour
