The manager of a warehouse must decide on the number of loading docks to request fo a new facility in order to minimize the sum of dock costs and driver-truck costs.

The manager has learned that each driver-truck combination represents a cost of $600 per day and that each dock plus loading crew represents a cost of $1,000 per day.

(a) How many docks should be requested if trucks arrive at the rate of three per day, each dock can handle five trucks per day, and both rates are Poisson? Show support to your answer.

(b) An employee has proposed adding new equipment that would speed up the loading rate to 5.71 per day. The equipment would cost $100 per day for each dock.

Should the manager invest in the new equipment? Show support to your response.

Copy and paste onto this worksheet the output used in anwering the questions above.

When copying and pasting, use Copy > Paste Special > Picture

Hint: For parts (a) and (b), compute the cost of the servers (per day) and the cost associated

with trucks and drivers in the system per day.

See Example 6 on page 810 for a similar situation

Multiple Channel Waiting Line Model

Arrival rate l =

3

Service rate m =

5

Multiple Channel Waiting Line Model

Increment Dl =

0.1

Increment Dm =

0.1

Arrival rate l =

3

Service rate m =

5.71

Interarrival Time 1/l =

0.3333

Service time 1/m =

0.2000

Increment Dl =

0.1

Increment Dm =

0.1

Interarrival Time 1/l =

0.3333

Service time 1/m =

0.1751

Number of servers (max 12)

M =

1

2

3

4

5

6

System Utilization

r =

0.6000

0.3000

0.2000

0.1500

0.1200

0.1000

Number of servers (max 12)

M =

1

2

3

4

5

6

Probability system is empty

P₀ =

0.4000

0.5385

0.5479

0.5487

0.5488

0.5488

System Utilization

r =

0.5254

0.2627

0.1751

0.1313

0.1051

0.0876

Probability arrival must wait

P_w =

0.6000

0.1385

0.0247

0.0035

0.0004

0.0000

Probability system is empty

P₀ =

0.4746

0.5839

0.5908

0.5913

0.5913

0.5913

Average number in line

L_q =

0.9000

0.0593

0.0062

0.0006

0.0001

0.0000

Probability arrival must wait

P_w =

0.5254

0.1093

0.0173

0.0022

0.0002

0.0000

Average number in system

L_s =

1.5000

0.6593

0.6062

0.6006

0.6001

0.6000

Average number in line

L_q =

0.5816

0.0389

0.0037

0.0003

0.0000

0.0000

Average time in line

W_q =

0.3000

0.0198

0.0021

0.0002

0.0000

0.0000

Average number in system

L_s =

1.1070

0.5643

0.5291

0.5257

0.5254

0.5254

Average time in system

W_s =

0.5000

0.2198

0.2021

0.2002

0.2000

0.2000

Average time in line

W_q =

0.1939

0.0130

0.0012

0.0001

0.0000

0.0000

Average waiting time

W_a =

0.5000

0.1429

0.0833

0.0588

0.0455

0.0370

Average time in system

W_s =

0.3690

0.1881

0.1764

0.1752

0.1751

0.1751

Average waiting time

W_a =

0.3690

0.1188

0.0708

0.0504

0.0391

0.0320