someone essay

Midterm Formula Sheet 1. Process Analysis Little’s Law: Average Inventory = Average Flow Rate x Average Flow Time

Utilization = Flow Rate / Capacity = Actual Output Rate / Maximum Output Rate

Load Factor = Demand / Capacity = Demand Rate / Maximum Output Rate

2. General Concepts Coefficient of Variation (of a random variable or distribution) = Standard Deviation / Mean

3. Queueing – Notation & Terminology Number of Servers: 𝑘 Coefficient of Variation for Inter-Arrival Times: 𝐶𝑉𝑎 Arrival Rate: 𝜆 Coefficient of Variation for Processing (Service) Times: 𝐶𝑉𝑝

Service Rate (per server): 𝜇 𝑎 = Inter-Arrival Time = 1 / Arrival Rate = 1/𝜆 Utilization: 𝜌 𝑝 = Activity Time = 1 / Service Rate = 1/𝜇

4. Queueing – M/M/1 Queue (Single Server, Poisson Arrivals, Exponential Service Times) 𝜌 = Utilization = 𝜆/𝜇 𝑇𝑞 = Average time in queue = 𝜌/(𝜇 − 𝜆)

𝐼 = Average number in system = 𝜆/(𝜇 − 𝜆) 𝑃0 = Prob(system is empty) = 1 − 𝜌 𝐼𝑞 = Average number in queue = 𝜌𝜆/(𝜇 − 𝜆) 𝑃𝑛 = Prob(𝒏 customers in system) = (1 − 𝜌)𝜌

𝑛

𝑇 = Average time in system = 1/(𝜇 − 𝜆)

5. Queueing – G/G/1 Queue (Single Server, General Arrivals, General Service Times)

𝑇𝑞 = E[Time in Queue] = Activity Time × ( utilization

1 − utilization ) × (

𝐶𝑉𝑎 2 + 𝐶𝑉𝑝

2

2 )

6. Queueing – G/G/k Queue (Multiple Servers, General Arrivals, General Service Times)

𝑇𝑞 = E[Time in Queue] = ( Activity Time

𝑘 ) × (

utilization√2(𝑘+1)−1

1 − utilization ) × (

𝐶𝑉𝑎 2 + 𝐶𝑉𝑝

2

2 )

7. Queueing – Formulas That Apply to All Queues (M/M/1, M/G/k, G/G/1, G/G/k) 𝜌 = Utilization = 𝜆/(𝑘𝜇) 𝐼𝑝 = Inventory in Service = 𝑘𝜌

𝑇 = Average Flow Time = 𝑇𝑞 + 1/𝜇 𝐼𝑞 = Inventory in the Queue = 𝜆𝑇𝑞

𝐼 = Inventory in the System = 𝐼𝑝 + 𝐼𝑞 = 𝜆𝑇

8. Queueing – Throughput Loss (Multiple Servers, Poisson Arrivals, General Service Times)

Erlang Loss Function: 𝑃𝑘 (𝑟); know how to look this up in a table (to be provided on exam)

where 𝑟 = 𝑝/𝑎 = Activity Time / Inter-Arrival Time = 𝜆/𝜇 = Arrival Rate / Service Rate