Simulation of Queuing problems using Random numbers

-- Renuka Narang

Simulation

Simulation is imitation of some real thing, or a process.

The act of simulating something generally involves representation of certain key characteristics or behaviours

of a selected physical or abstract system. Simulation involves the use of models to represent real

life situation.

Simulation Model

A simulation model is a mathematical model that calculates the impact of uncertain inputs and decisions we make on outcomes that we care about, such as profit and loss, investment returns, etc.

A simulation model will include:

Model inputs that are uncertain numbers/ uncertain variables

Intermediate calculations as required Model outputs that depend on the inputs -- These

are uncertain functions

Simulation techniques

Simulation techniques can be used to assist management decision-making, where analytical methods are either not available or inappropriate.

Typical business problems where simulation could be used to aid management decision-making are Inventory control. Queuing problems. Production planning.

Simulation and Queuing problems.

A major application of simulation has been in the analysis of waiting line, or queuing systems.

Since the time spent by people and things waiting in line is a valuable resource, the reduction of waiting time is an important aspect of operations management.

Waiting time has also become more important because of the increased emphasis on quality. Customers equate quality service with quick service and providing quick service has become an important aspect of quality service

Queuing problems.

For queuing systems, it is usually not possible to develop analytical formulas, and simulation is often the only means of analysis.

Simulation can hence be used to investigate problems that are common in any situation involving customers, items or orders arriving at a given point, and being processed in a specified order.

For ex: Customers arrive in a bank and form a single queue,

which feeds a number of service desks. The arrival rate of the customers will determine the number of service desks to have open at any specific point in time

Components of queuing systems

A queue system can be divided into four components Arrivals: Concerned with how items (people, cars etc)

arrive in the system. Queue or waiting line: Concerned with what happens

between the arrival of an item requiring service and the time when service is carried out.

Service: Concerned with the time taken to serve a customer.

Outlet or departure: The exit from the system. A queuing problem involves striking a balance

between the cost of making reductions in service time and the benefits gained from such a reduction

Structures of queuing system

There are a number of structures of queuing systems in practice.

We will study only one i.e. single queue – single service point. Single queue – single service point Queue discipline is first come – first served. Arrivals* are random and for simulation this

randomness must be taken into account. Service times** are random and for simulation this

randomness must be taken into account *Inter-arrival time: Is the time between the arrival of successive

customers in a queuing situation.**Service time: Is the length of time taken to serve customers

Random Numbers

What is the purpose of random numbers? There is randomness in the way customers are

likely to arrive. The service time in most of the cases is also

variable. The purpose of the random numbers is to allow

you to randomly select an arrival or service time from the appropriate distribution.

To account for randomness, random numbers are used.

Random Numbers

Such numbers can be computer generated, and are often listed in published statistical tables.

Here we have a set of random numbers.89 07 37 29 28 08 75 01 21 6334 65 11 80 34 14 92 48 83 91 52 49 98 44 80 04 42 37 87 96

The random numbers are displaced as two-digit numbers in the range between 00 and 99.

Every number is equally likely to occur and there is no pattern, and thus no way of predicting what number will be next in the sequence.

Example Problem

The arrival time of a customer at a retail sales depot is according to the following distribution

Simulate the process for 10 arrivals and estimate the average waiting time for the customer and percentage idle time for the server.

Use the following random numbers: For IAT: 25, 19, 64, 82, 62, 74, 29, 92, 24, 23, 68, 96. For ST: 92, 41,66,07,44,29,52,43,87,55,47,83 Assume that the shop opens at 9:00 am in the morning.

Inter-arrival timeProbability Service time

Probability(in minutes) (in minutes)

3 0.1 3 0.34 0.2 4 0.55 0.5 5 0.16 0.1 6 0.17 0.1

Solution

Inter arrival time Probability Cumulative Probability Basis of random allocation

3 0.1 0.1 0.0 -- 0.094 0.2 0.3 0.1 -- 0.295 0.5 0.8 0.3 -- 0.796 0.1 0.9 0.9 -- 0.897 0.1 1 0.9 -- 0.99

Service time Probability Cumulative Probability Basis of random allocation

3 0.3 0.3 0.0 -- 0.294 0.5 0.8 0.3 -- 0.795 0.1 0.9 0.8 -- 0.896 0.1 1 0.9 -- 0.99

Calculation of Basis of random allocation

Solution

Customer Random Number

Inter arrival time

Random number

Service time

Time of arrival

Service starts at

Service ends at

Waiting customer Idle time

Inter arrival time

34567

Service time

3456

Basis of random allocation0.0 -- 0.090.1 -- 0.290.3 -- 0.790.9 -- 0.890.9 -- 0.99

Basis of random allocation

0.0 -- 0.290.3 -- 0.790.8 -- 0.890.9 -- 0.99

1 25 4 92 6 9.04 9.04 9.10 -- 4

2 19 4 41 4 9.08 9.10 9.14 2 --

3 64 5 66 4 9.13 9.14 9.18 1 --

4 82 6 07 3 9.19 9.19 9.22 -- 1

5 62 5 44 4 9.24 9.24 9.28 -- 2

9.29 9.29 9.32 -- 1

9.33 9.33 9.37 -- 1

9.40 9.40 9.44 -- 3

9.44 9.44 9.49 -- --

9.48 9.49 9.53 1 --

Total time 53 minsTotal waiting time 4 minsTotal idle time 12 mins

6 74 5 29 3

7 29 4 52 4

8 92 7 43 4

9 24 4 87 5

10 23 4 55 4

Solution

Average waiting time per customer is 4/10 = .4 minutes

Percentage average for the server is (12/53)*100 = 22.64%

Waiting time should be as less as possible!!

Thank you!