Grocery Stores

Simulation of Queueing Systems

A queuing system is described by

  • Calling population
  • Arrival rate
  • Service mechanism
  • System capacity
  • Queueing discipline

Example

  • Analysis of a small grocery store

    • One checkout counter
    • Customers arrive at random times from {1,2,…,8}
    • Service times vary from {1,2,…,6}
    • Consider the system for 100 customers

  • Problems/Simplifications

    • Sample size is too small to be able to draw reliable conclusions
    • Initial condition is not considered

  • Interesting results for a manager, but

    • longer simulation run would increase the accuracy

  • Some interpretations

    • Average waiting time is not high
    • Server has not undue amount of idle time, it is well loaded
    • Nearly half of the customers have to wait

Final Exam Question (May 2018)

A queuing system in a minimarket occur randomly with inter-arrival time {1..6} and service time {1..10}. The following table Q1 depicts results of the simulation towards five regular customers.

Customer No Customer Name Inter-arrival time Arrival Time on clock Service time
1 Bobby - 0 10
2 Lucy 5 5 8
3 Anie 2 7 9
4 Diana 1 8 5
5 Stewart 2 10 2

Analyze the simulation results and answer the following questions with appropriate explanation:

Find ONE(1) potential complain that may arise from the above simulation. What is the best way to avoid this complain?

ss

Were Bobby and Lucy queuing before approaching cashier? What could be the reason?

How long the cashier were idle in between services? Will it affect the business?

The cashier were not idle in between services.

Assume Diana is a customer who uses a wheelchairand she comes alone. Should she deserve a special assistance while queuing? Discuss your answer.

Customer Interarrival Time Arrival Time Service Time Time Service Begins Time Service Ends Waiting Time in Queue Time Customer in System Idle Time of Server
1 0 0 10 0 10 0 10 0
2 5 5 8 10 18 5 13 0
3 2 7 9 18 27 11 20 0
4 1 8 5 27 32 19 24 0
5 2 10 2 32 34 22 24 0
Total 10 34 57 91 0

Call Centers

  • Consider a Call Center where technical personnel take calls and provide service

  • Two technical support people (2 server) exists

    • Able - more experienced, provides service faster
    • Baker - provides service slower

  • Rule

    • Able gets call if both people are idle
    • Try other rules
      • Baker gets call if both are idle
      • Call is assigned randomly to Able and Baker

  • Goal of study: Find out how well the current rule works

  • Interarrival distribution of calls for technical support

  • Simulation run for 100 calls

Tutorial

Inter-Arrival Time (I) and Service Time (S) is depicted as {I,S} below.
{0,3}
{2,4}
{3,2}
{2,5}
{2,2}
{3,3}

Identify how many calls completed by Able and Baker, caller delay and time in the system for the above arrivals.

Caller Nr. Interarrival Time Arrival Time When Able Avail. When Baker Avail. Server Chosen Service Time Time Service Begins Able’s Service Compl. Time Baker’s Service Compl. Time Caller Delay Time in System
1 - 0 0 0 Able 3 0 3 0 3
2 2 2 3 2 Baker 4 2 6 0 4
3 3 5 5 6 Able 2 5 7 0 2
4 2 7 7 7 Able 5 7 12 0 5
5 2 9 12 9 Baker 2 9 11 0 2
6 3 12 12 11 Able 3 12 15 0 3
Total 0 19

Inventory System

  • Distributes items from current inventory to customers
  • Customer demand is discrete
  • Simple <-> one type of item

Final Exam Question

Question 1

An inventory system is doing a periodic inventory review and simulation to ensure the flow is balanced between stock and demand. Let (s,S) = (25,65) where s is the minimum inventory level and S is the maximum inventory level.

Discuss FIVE(5) inventory system costs that affect the performance of an inventory system.

  • Holding cost
    for items in inventory
  • Shortage cost
    for unmet demand
  • Setup cost
    fixed cost when order is placed
  • Item cost
    per-item order cost
  • Ordering cost
    sum of setup and items costs

Assume n = 10 for the time intervals and the value of i (inventory level) and di (demand quantity during the interval) is given by the following table.

i 1 2 3 4 5 6 7 8 9 10
di 40 25 35 25 55 40 35 25 30 45

With the aid of the diagram, find any shortage that occurs in the system during the interval. Should the inventory level (s,S) be modified based on this periodic review? Explain your answer.

Poisson Distribution

  • A Poisson distribution helps in describing the chances
    of occurrence of a number of events in some given time interval that the value of average number of
    occurrence of the event is known.
  • This is a major and only condition of Poisson distribution.
  • An experiment in statistics is termed as Poisson experiment when it possesses the following probabilities:
    • The outcomes of the experiment can be easily classified as either success or failure.
    • The average of the number of successes within a region that is specified is known.
    • The probability of occurrence of a success is always proportional to the size of the specified region.
    • The probability of occurrence of success in a very small region is zero virtually.
    • It is to be noted that the region that is specified can take different forms like area, length, time period etc

Predict future occurences based on history
Step 1 - get &#x03BB (history of things occured)
Step 2 - get table of &#x03BB (poisson distribution)
Step 3 - project probability numbers ~ exact value ~ range of numbers

Example

x &#x03BB P(x) for &#x03BB = 3
0 3 0.04979
1 3 0.14936
2 3 0.22404
3 3 0.22404
4 3 0.16803
5 3 0.10082
6 3 0.05041
7 3 0.02160
8 3 0.00810
9 3 0.00270
10 3 0.00081
11 3 0.00022
12 3 0.00006
13 3 0.00001
14 3 0.00000
15 3 0.00000
  1. A man was able to complete 3 files a day on an average. Find the probability that he can complete 5 files the next day. Hint: find the number on the table.

P(X = 5) = 0.00757

  1. The number of calls coming per minute into a hotels reservation center is Poisson random variable with mean. Find the probability that no calls come in a given 1 minute period. Hint: find the probability number on the table.

P(X = 0) = 0.00005

  1. Let X equal the number of typos on a printed page with a mean of 3 typos per page. What is the probability that a randomly selected page has at least one typo on it? Hint: P(X ≥ 1) = 1 − P(X = 0)

P(X ≥ 1) = 1 − P(X = 0) = 1 - 0.00005 = 0.99995

Monte Carlo Simulation

Predict future achievement of a company based on random numbers
Step 1 - relative frequency table
Step 2 - range of distribution
Step 3 - random numbers are projected to the range of distribution
Step 4 - Calculate the company revenue