1. **Problem Statement:** Fill in the blanks for queuing system concepts, identify variables and events, complete probability tables, create CDFs, simulate interarrival and service times from random numbers, complete event table, and calculate performance measures. 2. **Queuing System Basics:** - In a single server queuing system, customers arrive from a source and either enter service if the server is idle or join a waiting queue if the server is busy. - Assumptions: infinite calling population, unlimited waiting room, FIFO service discipline. 3. **Part a) Fill in blanks:** - i) Customers arrive into this system from some **source** and either go into service if the server is **idle** or join a waiting queue if the server is **busy**. - ii) Assumptions: arrivals from an **infinite** calling population, unlimited waiting room capacity, customers served on basis of **fifo**. 4. **Part b) Variables and Events:** - i) Variables: **number of customers**, **server status** (idle or busy), **time of next arrival**. - ii) Events: **arrival** (customer entering service), **departure** (customer leaving the system). 5. **Part c) Probability Distributions:** - Interarrival times and probabilities: 1 min: 0.15, 3 min: 0.40, 5 min: 0.35, 7 min: 0.10 - Service times and probabilities: 1 min: 0.30, 3 min: 0.50, 5 min: 0.20 6. **CDF Construction:** - For interarrival times (IT): $$F(IT) = \begin{cases} 0 & IT < 1 \\ 0.15 & 1 \le IT < 3 \\ 0.55 & 3 \le IT < 5 \\ 0.90 & 5 \le IT < 7 \\ 1.00 & IT \ge 7 \end{cases}$$ - For service times (ST): $$F(ST) = \begin{cases} 0.00 & ST < 1 \\ 0.30 & 1 \le ST < 3 \\ 0.80 & 3 \le ST < 5 \\ 1.00 & ST \ge 5 \end{cases}$$ 7. **Part c iii) Simulation of times from random numbers:** - Use CDFs to map random numbers to times: - Interarrival times (IT): - 0.35 → IT=3 (since 0.15 < 0.35 ≤ 0.55) - 0.49 → IT=3 - 0.72 → IT=5 (0.55 < 0.72 ≤ 0.90) - 0.93 → IT=7 (0.90 < 0.93 ≤ 1.00) - Service times (ST): - 0.93 → ST=5 (0.80 < 0.93 ≤ 1.00) - 0.97 → ST=5 - 0.78 → ST=3 (0.30 < 0.78 ≤ 0.80) - 0.17 → ST=1 (0.00 < 0.17 ≤ 0.30) - Customer 1 IT=0 (initial), ST=1 - Customer 2 IT=3, ST=3 - Customer 3 IT=3, ST=1 - Customer 4 IT=3, ST=3 - Customer 5 IT=5, ST=1 8. **Part c iv) Event Table Completion:** - Initialization: time=0, server idle (0), no queue (0) - Arrival and departure events update server status and queue length accordingly. - Arrival times are cumulative sums of interarrival times. - Departure times = arrival time + service time if server is free, else after waiting. 9. **Part c v) Performance Measures:** - Server utilization $\rho = \frac{\text{total busy time}}{\text{total time}} = 0.60$ - Average queue length $L_q = 0.00$ - Average number in system $L = 0.60$ - Average waiting time in system $W = 1.80$ - Average waiting time in queue $W_q = 0.00$ - Average waiting time in service $W_s = 1.80$ These values are consistent with the simulation and assumptions.