Operasyon Yönetim Bekleme Hattı Modelleri

Outline BEKLEME HATTI SİSTEMİNİN ÖZELLİKLERİ Geliş özellikleri Bekleme hattı özellikleriWaiting-Line Characteristics Hizmet tesisi Özellikleri Kuyruk Performansının Ölçülmesi Bekleme maliyetleri Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Outline THE VARIETY OF QUEUING MODELS OTHER QUEUING APPROACHES Model A: Single-Channel Queuing Model with Poisson Arrivals and Exponential Service Times Model B: Multiple-Channel Queuing Model Model C: Constant Service Time Model Model D: Limited Population Model OTHER QUEUING APPROACHES Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Learning Objectives When you complete this chapter, you should be able to Identify or Define: The assumptions of the four basic waiting-line models Explain or be able to use: How to apply waiting-line models How to conduct an economic analysis of queues Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

You’ve Been There Before! © 1995 Corel Corp. Thank you for holding. Hello...are you there? ‘The other line always moves faster.’ ‘If you change lines, the one you left will start to move faster than the one you’re in.’ Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Waiting Line Examples Situation Arrivals Servers Service Process Bank Customers Teller Deposit etc. Doctor’s Patient Doctor Treatment office Traffic Cars Light Controlled intersection passage Assembly line Parts Workers Assembly Tool crib Workers Clerks Check out/in tools Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Waiting Lines First studied by A. K. Erlang in 1913 Analyzed telephone facilities Body of knowledge called queuing theory Queue is another name for waiting line Decision problem Balance cost of providing good service with cost of customers waiting Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Total waiting line cost Waiting Line Costs Cost Total waiting line cost Service cost Waiting time cost Level of service Optimal Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Waiting Line Terminology Queue: Waiting line Arrival: 1 person, machine, part, etc. that arrives and demands service Queue discipline: Rules for determining the order that arrivals receive service Channel: Number of waiting lines Phase: Number of steps in service Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Three Parts of a Queuing System at Dave’s Car-Wash Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Characteristics of a Waiting Line System Service Facility Waiting Line Population Arrival rate distribution Poisson Other Pattern of arrivals Random Scheduled Arrival Characteristics Size of the source population Limited Unlimited Behavior of the arrivals Join the queue, and wait until served Balk; refuse to join the line Renege; leave the line Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Characteristics of a Waiting Line System - Continued Service Facility Waiting Line Population Waiting Line Characteristics Length of the queue limited unlimited Service priority FIFO other Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Characteristics of a Waiting Line System - Continued Service Facility Waiting Line Population Service Facility Characteristics Number of channels single multiple Number of phases in service system Service time distribution negative exponential other Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Waiting Line System Service system Waiting line Service facility Input source © 1995 Corel Corp. Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Input Characteristics Input Source (Population) Size Infinite Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Input Characteristics Input Source (Population) Size © 1995 Corel Corp. Fixed number of aircraft to service Infinite Finite Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Input Characteristics Input Source (Population) Size Arrival Pattern Finite Infinite Random Non- Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Input Characteristics Input Source (Population) Size Arrival Pattern Finite Infinite Random Non- Poisson Other Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Poisson Dağılımı Belli bir zaman diliminde gerçekleşen olay sayısı Örneğin: 15 dakika içinde gelen müşteri sayısı. Ortalama=  (e.g., 5/saat.) Olasılık:  = 0.5  = 6.0 Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Geliş Zamanları için Poisson Dağılımı Olasılık Olasılık =2 =4 Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Girdi Özellikleri Girdi Kaynağı (Nüfus) Büyüklük Davranış Geliş Tarzı Sonlu Sonsuz Rassal değil Sabırlı Sabırsız Poisson Diğer Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Input Characteristics Input Source (Population) Size Behavior Arrival Pattern Finite Infinite Random Non- Patient Impatient Girmeden terkediyor Poisson Other Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Balking Input source Service facility Waiting line Service system © 1995 Corel Corp. Line was too long! Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 2

Input Characteristics Input Source (Population) Size Behavior Arrival Pattern Finite Infinite Random Non- Patient Impatient Balk Sonra terkediyor Poisson Other Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Terk Etme Input source Service facility Waiting line Service system © 1995 Corel Corp. Vaz geçtim Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 2

Waiting Line Characteristics Uzunluk Sınırsız © 1995 Corel Corp. Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Waiting Line Characteristics Length Sınırlı Unlimited © 1995 Corel Corp. Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Waiting Line Characteristics Length Queue Discipline Limited Unlimited İGİÇ (İGİH Rassal Öncelik Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Service Characteristics Facility Korfigurasyon Çok Kanallı Tek kanallı aşamalı Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Negatif Üstel Dağılım Sürekli dağılım. Olasılık: Örnek: Gelişler arasındaki süres. Ortalama hizmet hızı =  6 müşteri/saat. Ortalama hizmet süresi süresi = 1/ 1/6 saat = 10dakika Probability t>x =1 =2 =3 =4 D-30 30

Temel Modelin Varsayımları Müşteri nüfusu homojen ve sonsuzdur. Kuyruk kapasitesi sonsuzdur. Müşteriler iyi davranmaktadır (kuyruğa girmeden veya girdikten sonra terk yok). Gelenler İGİH alıyor İGİH (FIFO). Poisson gelişler. Gelişler arasındaki süre negatif Poisson dağılımına uygun Hizmet süreleri negatif üstel dağılımla tanımlanıyor D-31 31

Hizmet Süreleri Rassal: Negatif üstel olasılık dağılımını kullan. Mean service rate =  6 customers/hr. Mean service time = 1/ 1/6 hour = 10 minutes. Rassal değil: Sabit olabilir. Örnek: Otomatik arasa yıkama D-32 32

Negatif Üstel Dağılım Ortalama hizmet süresi= 1 saat Ortalama hizmet süresi=20 dakika Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Single-Channel, Single-Phase System Arrivals Served units Service facility Queue Service system Dock Waiting ship line Ships at sea Ship unloading system Empty ships Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 63

Single-Channel, Multi-Phase System Arrivals Served units Service facility Queue Service system Pick-up Waiting cars Cars in area McDonald’s drive-through Pay Cars & food Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 64

Multi-Channel, Single Phase System Arrivals Served units Service facility Queue Service system Example: Bank customers wait in single line for one of several tellers. Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 62

Multi-Channel, Multi-Phase System Service facility Arrivals Served units Queue Service system Example: At a laundromat, customers use one of several washers, then one of several dryers. Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 63

Two Examples of the Negative Exponential Distribution Average Service Rate () = 3 customers per hour Average Service Time = 20 minutes per customer Average Service Rate () = 1 customer per hour Probability that Service Time is greater than t=e-t, for t > 0 Time (t) in Hours Probability that Service Time  t Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Optimum Hizmet Düzeyine Karar Verme Beklenen toplam mal. Bekleme mal. Maliyet Düşük hizmet düzeyi Optimal hizmet düzeyi Yüksek hizmet düzeyi Minimum toplam mal. Hizmet verme mal. Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Waiting-Line Performance Measures Average queue time, Wq Average queue length, Lq Average time in system, Ws Average number in system, Ls Probability of idle service facility, P0 System utilization,  Probability of k units in system, Pn > k Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Assumptions of the Basic Simple Queuing Model Arrivals are served on a first come, first served basis Arrivals are independent of preceding arrivals Arrival rates are described by the Poisson probability distribution, and customers come from a very large population Service times vary from one customer to another, and are independent of one and other; the average service time is known Service times are described by the negative exponential probability distribution The service rate is greater than the arrival rate Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Types of Queuing Models Basit (M/M/1) Example: AVM’de enformasyon kulübesi Çok kanallı (M/M/S) Example: Uçak bilet kuyruğu Sabit hizmet (M/D/1) Example: Otomatik araba yıkama Sınırlı Nüfus Example: yalnızca 7 matkabı olan department Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Simple (M/M/1) Model Characteristics Type: Single-channel, single-phase system Input source: Infinite; no balks, no reneging Arrival distribution: Poisson Queue: Unlimited; single line Queue discipline: FIFO (FCFS) Service distribution: Negative exponential Relationship: Independent service & arrival Service rate > arrival rate Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Bekleme Modeli Notasyonu a/b/S Hizmet sunucu veya kanal sayısı. Hizmet süresi dağılımı. Geliş süresi dağılımı. M = Negatif üstel dağılım (Poisson gelişler). G = Genel dağılım. D = Deterministik (planlanmış). D-44 44

Basit (M/M/1) Model Denklemleri Kuyrukta ortalama birim sayısı Sistemde ortalama süre Kuyruktaki ortalama birim sayısı Kuyruktaki ortalama süre Ssistem kullanım oranı L W s q =   -  1 2  ( -  )   Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Basit (M/M/1) Olasılık dentlemleri Sistemde 0 birim olma olasılığı, yani sistem boş Sistemde k birimden fazla birim olma olasılığı n sistemdeki birim sayısıdır. Bu aynı zamanda sistemde k+1 veya daha fazla bulunma olasılığıdır P k+1 1 = - n>k    l ( ) Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Multichannel (M/M/S) Model Characteristics Type: Multichannel system Input source: Infinite; no balks, no reneging Arrival distribution: Poisson Queue: Unlimited; multiple lines Queue discipline: FIFO (FCFS) Service distribution: Negative exponential Relationship: Independent service & arrival  Service rates > arrival rate Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

M/M/1 Örnek1 Ortalama geliş (varış) hızı saatte 10’dur. Ortalama hizmet süresi 5 dakikadır.  = 10/hr and  = 12/hr (1/ = 5 dakika = 1/12 hour) S1: Ayrılışlar arasındaki ortalama süre nedir? 5 minutes? 6 minutes? S2: sistemdeki ortalama bekleme ne kadardır? 1 W = = 0.5 hour or 30 minutes s 12/hr-10/hr D-48 48

M/M/1 Örnek 1 Q3: Kuyruktaki ortalama bekleme nedir?  = 10/hr and  = 12/hr Q3: Kuyruktaki ortalama bekleme nedir? 10 W = O.41667 hours = 25 minutes = q 12 (12-10) Also note: so W s = 1  q + - 2 12 O.41667 hours D-49 49

M/M/1 Example 1  = 10/hr and  = 12/hr Q4: Kuyruktaki ve sistemdeki ortalama müşteri sayısı nedir? 102 = 4.1667 müşteri L = q 12 (12-10) 10 5 customers L = = s 12-10 Also note: L q =  W = 10  0.41667 = 4.1667 q L s =  W = 10  0.5 = 5 s D-50 50

M/M/1 Example 1  = 10/hr and  = 12/hr Q5: sistemin boş olduğu zaman yüzdesi nedir (hizmet veren boş duruyor)? 1 -   10 P = 1 -  = = 1 - = zamanın 16.67%’si 12 Q6: Sistemde 5 müşteriden fazla bulunma olasılığı? ( ) 6 10 P = = Zamanın 33.5%’i n>5 12 D-51 51

Sistemde 5’ten fazla bulunması Sistemde 5’ten fazla bulunma şunlarla aynıdır: “kuyrukta 4’ten fazla” “kuyrukta 5 veya daha fazla” “sistemde 6 veya daha fazla”. Hepsi P n>5 D-52 52

M/M/1 Example 1  = 10/hr and  = 12/hr Q7: Günde (8 saatte) kuyrukta 5 veya daha fazla bulunan zaman dilimi? P = 0.335 so 33.5% of time there are 6 or more in line. n>5 0.335 x 480 min./day = 160.8 min. = ~2 hr 40 min. Q8: Kuyrukta 3 veya daha az olduğu zaman yüzdesi? 5 = 1 - 1 - P n>4 ( ) 10 12 = 1 - 0.402 = 0.598 or 59.8% D-53 53

M/M/1 Example 2 8 saatlik günde ortalama beş fotokopi makinası bozulmaktadır. Ortalama servis süresi ir saat 15 dakikadır.  = 5/day ( = 0.625/hour) 1/ = 1.25 hours = 0.15625 days  = 1 every 1.25 hours = 6.4/day Q1: Sistemde bekleyen ortalama “müşteri” sayısı nedir? 5/day L = = 3.57 bozulan makina S 6.4/day-5/day D-54 54

M/M/1 Example 2 Q2: Kuyrukta ortalama bekleme ne kadardır?  = 5/day (or  = 0.625/hour)  = 6.4/day (or  = 0.8/hour) Q2: Kuyrukta ortalama bekleme ne kadardır? 5 W = = 0.558 days (or 4.46 hours) q 6.4(6.4 - 5) 0.625 W = = 4.46 hours q 0.8(0.8 - 0.625) D-55 55

M/M/1 Example 2  = 5/day (or  = 0.625/hour)  = 6.4/day (or  = 0.8/hour) Q3: Günde tamir personeli için bekleyen makine sayısının 2 veya daha fazla olduğu süre ne kadardır?How much time per day (on average) are there 2 or more broken copiers waiting for the repair person? “Kuyrukta” 2 veya daha fazla= sistemde 2’den fazla ( ) 3 5 P = = 0.477 (zamanın 47.7%’si) n>2 6.4 0.477x 480 min./day = 229 min. = 3 hr 49 min. D-56 56

Model B (M/M/S) Equations Probability of zero people or units in the system: Average number of people or units in the system: Average time a unit spends in the system: Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Model B (M/M/S) Equations Average number of people or units waiting for service: Average time a person or unit spends in the queue Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Constant Service Rate (M/D/1) Model Characteristics Type: Single-channel, single-phase system Input source: Infinite; no balks, no reneging Arrival distribution: Poisson Queue: Unlimited; single line Queue discipline: FIFO (FCFS) Service distribution: Negative exponential Relationship: Independent service & arrival  Service rates > arrival rate Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Model C (M/D/1) Equations Average number of people or units in the system: Average time a unit spends in the system: Average number of people or units waiting for service: Average time a person or unit spends in the queue Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Limited Population Model (D) Characteristics Type: Single-channel, single-phase system Input source: Limited; no balks, no reneging Arrival distribution: Poisson Queue: Limited; single line Queue discipline: FIFO (FCFS) Service distribution: Negative exponential Relationship: Independent service & arrival Service rate > arrival rate Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Model D (Limited Population) Equations Service Factor: Average number of people or units waiting for service: Average time a person or unit spends in the queue Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Model D (Limited Population) Equations - Continued Average number running Average number being served: Number in the population: Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Model D (Limited Population) Equations - Continued Where: D = probability that a unit will have to wait in the queue F = efficiency factor H = average number of units being serviced J = average number of units not in the queue or service bay L = average number of units waiting for service Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Model D (Limited Population) Equations - Continued M = number of service channels N = number of potential customers T = average service time U = average time between unit service requirements W = average time a unit waits in line X = service factor to be obtained from finite queuing tables Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Remember:  &  Are Rates  = Mean number of arrivals per time period e.g., 3 units/hour  = Mean number of people or items served per time period e.g., 4 units/hour 1/ = 15 minutes/unit If average service time is 15 minutes, then μ is 4 customers/hour © 1984-1994 T/Maker Co. Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458