Describe the queueing model

Question

Question 1: Problem Solving

Customers arrive at a cashier counter with a mean rate of 12 customers per hour. Based on the FCFS service discipline, a customer is served every 4 minutes on average and leaves the cashier counter. We assume that the cashier counter queuing system is an M/M/1 process in steady-state condition.

1. Describe the M/M/1 queueing model. What is the importance of the assumption above?

2. Find the cashier counter performance measures L, L_q, W, W_q.

Question 2: Problem Solving

Company operates a hydraulic pump which declines in productivity from over-heating. End of month inspection classifies pump state as: 0: Good as new; 1: Operable-Low overheating; 2: Operable-High overheating; 3: Inoperable-Failure.

Depending on the pump state, there are three possible decisions D₁, D₂, and D₃ the company manager can make, which are defined as follows:

D₁: Do nothing, applicable to states 0, 1, 2.

D₂: Pump Overhaul (takes pump back to state 1), applicable to state 2 only.

D₃: Replace pump with new (takes system back to state 0), applicable to states 2, 3.

The company manager has to choose between the three following maintenance policies (based on D₁, D₂, and D₃) to minimize cost:

R_a: D₁ in states 0, 1, 2; D₃ in state 3

R_b: D₁ in states 0, 1; D₂ in state 2; and D₃ in state 3

R_c: D₁ in states 0, 1; D₃ in states 2, 3

Assume that R_a, R_b, and R_c are stationary policies with respective transition matrices as follows:

1. Describe the Markov property in this case, and comment on the assumption made.

2. Based on the transition matrices above, determine the steady-state probabilities π₀ (state 0), π₁ (state 1), π₂ (state 2), and π₃ (state 3) for one of the three policies R_a, R_b, or R_c.

3. Now, if the operation costs according to the pump state are as follows:

Costs* (AED)

State	Lost production	Lost profit	Overhaul (Maintenance)	Replacement
0	0  (only D1)	0  (only D1)	0  (only D1)	0  (only D1)
1	1000  (only D1)	0  (only D1)	0  (only D1)	0  (only D1)
2	3000 for D1, 0 for D2 / D3	2000 for D2 / D3, 0 for D1	2500 for D2, 0 for D1 / D3	4000 for D3, 0 for D1 / D2
3	0  (only D3)	2000  (only D3)	0  (only D3)	4000  (only D3)

and the steady-state probabilities for the three policies are as follows:

steady-state probabilities
Policy	π0 (state 0)	π1 (state 1)	π2 (state 2)	π3 (state 3)
Ra	02-Sep	03-Sep	02-Sep	02-Sep
Rb	Feb-13	Jul-13	Feb-13	Feb-13
Rc	02-Jul	03-Jul	01-Jul	01-Jul

Find the optimal (minimum cost) policy based on the expected average operating cost per month.

Question 3: Problem Solving

Solve the following BIM problem by 0-1 Knapsack (Branch and Bound Algorithm)

Maximize z = 9x₁ + 3x₂ + 4x₃

subject to: 8x₁ + 3x₂ + 5x₃ ≤ 14

xj = 0 or 1, j = 1, 2, 3

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