Problems of queueing model

• Simulation examples• Single channel queue example• Able-baker example

• Inventory system

Able baker problem• In this model,• There are two servers. Able and baker• Able server does the job better than the Baker server• Baker gets the customer when the able server is busy•When both the servers are idle, able server gets the

customer

Example 1• Simulate the able-baker problem for 10 customers given that the

interarrival distribution and service time distribution is as given below.

• Calculate the following• Average waiting time• Average service time of able and baker server

• Consider the following random numbers for• Arrival time : 26, 98, 90, 26, 42, 74, 80, 68, 22• Service time : 95, 21, 51, 92, 89, 38, 13, 61, 50, 49

Inter arrival time Probability

1 0.25

2 0.40

3 0.20

4 0.15

Service time-able Probability

2 0.30

3 0.28

4 0.25

5 0.17

Service time-Baker Probability3 0.35

4 0.25

5 0.20

6 0.20

Example 2• Simulate the able-baker problem for 10 customers given that the interarrival

distribution and service time distribution is as given below.

• Calculate the following• Average waiting time• Average service time of able and baker server• Average time customer spends in the system• Average time between arrivals

• Consider the following random numbers for• Arrival time : 9, 60, 73, 35, 88, 10, 21, 49, 53• Service time :32, 94, 79, 5, 75, 84, 57, 55, 30, 50

Inter arrival time Probability

1 0.35

2 0.10

3 0.15

4 0.40

Service time-able Probability

2 0.4

3 0.1

4 0.3

5 0.2

Service time-Baker Probability3 0.35

4 0.20

5 0.25

6 0.20

Simulation of inventory system• Inventory system has a periodic review of length N at which

time the inventory level is checked.• An order is made to bring the inventory up to level M• At the end of review period an order of quantity Q is placed.• Demands are not known usually, so order quantities are

probabilities.• Demands are not usually uniform and do fluctuate over time

Example 1• A paper seller buys the paper for Rs 4 each and sells them for Rs 6 each, the

newspaper not sold at the end of day are sold as scrap for 0.5 each. There are three types of days “good, fair, poor” with probabilities 0.4, 0.35, 0.25 respectively. Develop a simulation table for purchase of 70 newspapers and demand for 10 days. Calculate the total profit. • Given that random numbers for types of news days are 94, 77, 49, 45 , 43,

37, 49, 0, 16, 24 and for demand are 80, 20, 15, 88, 98, 65, 86, 73, 24, 60. instead of 70 papers ,if 80 papers is purchased will it be more profitable?• Profit=revenue of sales- cost of newspaper-lost profit from excess demand +

salary of sales of scrap paper• Distribution of newspaper demand is as follows

Demand Good probability Fair probability Poor probability

40 0.03 0.10 0.44

50 0.05 0.18 0.22

60 0.15 0.40 0.16

70 0.20 0.20 0.12

80 0.35 0.08 0.06

90 0.15 0.04 0.00

100 0.07 0.00 0.00

Example 2• A baker bakes 30 dozens of bread each day. The probability distribution

of customers in table 1. customers order 1,2,3 or 4 dozens of bread loafs according to distribution given below in table 2. assume that each day all the customers order the same dozens of bread loafs. The selling price is rs 5.40 per dozen and making price is rs 3.80 per dozen. The left over bread loafs will be sold for half price at profit of baker. Instead of 30 dozens , if 40 dozens are baked per day will it be more profitable? • Random digits are • Customers : 50, 61, 73, 24, 96• Dozens : 5,3,7,0,8

Table 1Number of customers per day Probability

8 0.35

10 0.30

12 0.25

14 0.10

Table 2Number of dozens/customers Probability

1 0.4

2 0.3

3 0.2

4 0.1

Example 3• Dr XYZ is dentist who schedules all patients for 30 min

appointments. Some of the patients take more or less than 30 min. depending upon type of dental work to be done. The following table shows the various category of work, probability and time required to complete the work• Simulate the dental clinic for 3 hours and determine the

following• Average waiting time• Total idle time of the doctor

• Assume that patients show up at clinic at exactly at their scheduled time starting from 8.0 am. Use the following random numbers for handling the above problem• 40, 82, 11 , 34, 25, 66

Categories Probability Time requiredFilling 0.40 45

Crown 0.15 60

Cleaning 0.15 15

Extraction 0.10 45

Check up 0.20 15

Lead time inventory system

Example 1• Suppose that the maximum inventory level M is equal to 11

units and review period N in equal to 5 days.• The problem is to estimate the average ending units in the

inventory and number of days where shortage occurs for the problem lead time in random variable. Assume that the orders placed at the close of the business and received for the inventory at the beginning depending on the lead time. For this problem we begin inventory with 3 items and assume that the first order of 8 items arrive at third day morning

• Order quantity=order up to level-ending inventory + shortage quantity • Consider the following random numbers for different cycles

• For lead time : 5, 0, 3,4,8

Cycle Random digits1 24, 35, 65, 81, 542 3, 87, 27, 73, 703 47, 45, 48, 17, 094 42,87, 26, 36, 405 7, 63, 19, 88, 94

Random digit demands Probability

0 0.10

1 0.25

2 0.35

3 0.21

4 0.09

Random digit lead time Probability1 0.6

2 0.3

3 0.1

Example 2• Demand for widgets follows the following probability

distribution

• Stock is examined every 7 days (the plant is in operation every day) and if the stock level has reached 6 units or less an order for 10 widgets is placed. The lead time is probabilistic and follows the following distribution

Demand 0 1 2 3 4

Probability 0.33 0.25 0.20 0.12 0.10

• When the simulation begins 12 widgets are on hand and number of orders have back ordered (back ordering is allowed) • Simulate the operation of this system for 6 weeks. For lead time ,

random numbers are 3, 1, 1, 4, 0, 4• The random numbers for the simulation is

Lead time 0 1 2

Probability 0.3 0.5 0.2

Cycle Random digits1 94, 87, 63, 66, 30, 69, 372 01,66,51, 92, 36, 47, 803 94, 31,07, 09, 19, 29,294 94,87, 85, 66, 78, 94, 565 67, 07, 43, 36, 03, 46,166 74, 82, 31, 17, 00, 08, 85

Example 3• The number of fridge ordered each day is randomly

distributed as shown below

• The distribution of lead time is given below

Demand 0 1 2 3 4

Probability 0.10 0.25 0.35 0.21 0.09

Lead time 1 2 3Probability 0.6 0.3 0.1

• Assume that the orders are placed at the end of 5th day of each cycle. The simulation begins with inventory level at 3 refrigerators and an order of 8 refrigerators to arrive is 2 days. Simulate this for 5 cycles according up to level inventory system . Assume that order up to level is (m) is 11

End of unit 4Thank you