Waiting Lines: Queuing Theory PDF

MANAGEMENT SCIENCE Chapter 3 Waiting Lines: Queuing Theory Caballero, Irish Pangilinan, Jaypee Lagrosa, Richeene Petalver, Charlene Dane Paderes, Nep-Edralin Pidor, Meryll Joy Velasco, Beyonde Ira WAITING LINE WAITING LINE WAITING LINE WAITING LINE WAITING LINE Ord...

MANAGEMENT SCIENCE Chapter 3 Waiting Lines: Queuing Theory Caballero, Irish Pangilinan, Jaypee Lagrosa, Richeene Petalver, Charlene Dane Paderes, Nep-Edralin Pidor, Meryll Joy Velasco, Beyonde Ira WAITING LINE WAITING LINE WAITING LINE WAITING LINE WAITING LINE Ordered set of people, items or tasks set to be taken cared of based on a system of prioritization depending on the server’s policies and procedures Waiting line or Queue refers to an organized set of activities, elements work stations and people designed to handle and satisfy a queue of people, items or tasks. Queuing system  Why is there still waiting lines? Underloaded Server maybe a Large demand system & person or Irregularity of machine arrivals refers to mathematical approach to dealing with analysis of waiting lines Queuing theory minimize the amount of cost and time spent in the system while maximizing the number of entertained customers and eventually maximizing profits  POPULATION SOURCE - possible number of people that will pass through the system FINITE POPULATION SOURCE the server has a clear idea of the maximum number of people, items or customers that will go through the system INFINITE POPULATION SOURCE the population source is either unlimited or indeterminable  NUMBER OF SERVERS - number of channels (people or workstations) that can entertain customers Single Channel, Single Phase Single Channel, Multiple Phases Multiple Channels, Single Phase Multiple Channels, Multiple Phases  ARRIVAL PATTERNS - regularity or irregularity of intervals between the entry of people, items or tasks into the queuing system SERVICE PATTERNS - methodology used by the organization to entertain customers and the time spent entertaining each customer  QUEUE DISCIPLINE - the order of priority on how people, items or tasks are entertained FINITE POPULATION SOURCE NO FIRST IN, FIRST IN, PARTICULAR FIRST OUT LAST OUT ORDER  MEASURES OF SYSTEM PERFORMANCE The average number of customers waiting Performance Expectations Of Employees & Superiors The average time customers spend waiting in line and time spent during servicing System Utilization The implied cost of a given level of capacity and its related waiting line The probability that an arrival will have to wait for service INFINITE SOURCE Single Channel, Exponential Single Channel, Constant Service Time Service Time The server is human, The server is a machine, therefore performs therefore performs services at an average services at a constant rate. rate.  Symbols and Representation lamda λ Customer Arrival Rate myu µ Service Rate one over myu 1/µ Service Time ghe г Average number of customers being served at any given time rho ρ System Utilization  Symbols and Representation em М Number of servers or channels L sub q Lq Average number of customers waiting for service L sub s L­­s Average number of Customers in the system W sub q Wq Average time customers spend waiting in line W sub s Ws Average time customers spend in the system  Basic Relationships Certain basic relationships which are very helpful in deriving desired performance measures, given a few key values. System Utilization Average Number of Customers being Served at any given Time Average Number of Customers: Average Time of Customers: Percent of Idle Time Idle Time per hour  Basic Relationships Infinite-source model System ρ = _ λ_ Utilization Mµ Average Number of Customers being Served г = _ λ_ at any given Time µ  Basic Relationships Infinite-source model Average Number of Customers: Waiting in line for service: 2 Single Channel, Lq = ___λ___ Exponential Service Time µ(µ - λ) 2 Single Channel, Lq = ___λ___ Constant Service Time 2µ(µ - λ)  Basic Relationships Infinite-source model Average Number of Customers: In the System: L s = L­q + r Average Time of Customers: Waiting in Line In the System Wq = _L q_ Ws = Wq + _1_ _Ls _ or λ µ λ  Basic Relationships Infinite-source model Percent of Idle Time Percent Idle Time = 1 – p Idle Time per hour ( 1 – p ) x 60 minutes  Practice Problem Customers arrive at a bakery at an average rate of 18 customers per hour. Each clerk can serve a customer in an average of four minutes per service offering. Suppose that the average number of customers waiting in line is 3.6 people, determine the following. 1. Identify the arrival rate, service time and service rate. Arrival rate (λ) = 18 customers per hour Service time (1/µ) = 4 mins per customer Service rate (µ) = 15 customers per hour How to get: (4mins/60mins = 1/15) or (60mins/4mins = 15)  Practice Problem 2. Compute the average number of customers being served at any time. r = _λ_ = _18_ = 1.2 customers being served at any time µ 15 3. What is the average number of customers in the system? Given: Lq = 3.6 customers Ls = L q + r = 3.6 + 1.2 = 4.8 customers are in the system (waiting in line and being served)  Practice Problem 4. What is the average time that customers wait in line? W = _L _ = _3.6_ = 0.20 hours per customer, or 0.20 hours x λ 18 60 minutes per hour = 12 minutes 5. Approximately how long would it take a customer to go through the system? Ws can be determined using either of the following approaches: Ws = Wq + 1/µ = 0.20 hours + 1/15 hours = 0.267 hours or approximately 16 minutes Ws = Wq + 1/µ or (time waiting in line + service time) = 12 minutes (from letter d) + 4 minutes per customer (given) = 16 minutes  Practice Problem 6. Determine and interpret the system utilization for 2, 3 and 4 servers. p = __λ__ Mµ for M = 2 for M = 3 for M = 4 p = _18_ p = _18_ p = _18_ 2(15) 3(15) 4(15) = 0.60 = 0.40 = 0.30  Practice Problem 6. Determine and interpret the system utilization for 2, 3 and 4 servers. If the owner is tasked with choosing the ideal number of store clerks (from 2, 3 or 4 people) that will man the bakeshop, two is ideal number since it is the closest to 100% efficiency. Most people may think more people would be better, but expenses would also be greater. 0.60 or 60% means that the tow workers would be working 60% of the time (36 minutes every hour) and idle 40% of the time (14 minutes every hour). If the store will simultaneously use 4 people, they would be working 30% of the time (18 minutes per hour) and idle for 70% of the time (42 minutes per hour).  Management Science Thank You For Your Attention Group 2