Lognormal Model for Call-Center Service Times and Hazard Rate Functions

Transcription

1 Service Engineering - Recitatin 9 Lgnrmal Mdel fr Call-Center Service Times and Hazard Rate Functins Part 1. Service Times lgnrmal? (p.2-13) Part 2. Hazard Rate (p.14-18)

2 Part 1. Service Times lgnrmal? Review Basics f Lgnrmal Distributin 2

3 Part 1. Service Times lgnrmal? (2) Service times data Nvember and December service times (64704 calls) fr the fur majr service types: IN (5592), NE (7622), NW (5774) and PS (45716). Fr every service type, we check if the lgnrmal distributin fits Standard gdness-f-fit tests (chi-square, Klmgrv-Smirnv) reject the lgnrmal hypthesis. These tests are rarely applicable fr large samples f real data b/c the test recgnizes very small differences between real-data and theretical distributins. Hwever, the fit can be gd enugh fr applicatins. Hence, we use tw graphical tests, histgramsand Q-Q plts, t cmpare the sample service-time and lgnrmal distributins, and check if the differences are really significant fr ur purpses. 3

4 Part 1. Service Times lgnrmal? (3) Histgrams f Service Times versus Lgnrmal Density 1. transfrm the sample f service times via Ln transfrmatin (Ln(Service times)) 2. estimate μ and σ and use the frmulae n slide 3 t estimate the mean and standard deviatin f the lgnrmal distributin and get the lgnrmal cdf F 3. define T 0 =0, T 1, T 2, accrding t a chsen histgram bin size, and cmpute the empirical frequency fr each interval 4. fit lgnrmal distributin by calculating theretical prbabilities t fall int intervals P i =F(T i+1 )-F(T i ) and getting theretical frequencies by N i =N x P i. 5. cmpare with the histgram 4

5 Part 1. Service Times lgnrmal? (4) EX Histgrams f Service Times versus Lgnrmal Density bin size: 30 secnds, chsen by trial-and-errr The fit seems gd fr all service types IN: smewhat wrse, but nly in the middle f distributin. PS and verall are similar, but PS seems slightly better. gd fit at the tails fr IN and NE 5

6 Part 1. Service Times lgnrmal? (5) EX Histgrams f Service Times versus Lgnrmal Density 6

7 Part 1. Service Times lgnrmal? (6) EXHistgrams f Lg 10 (Service Times) versusnrmal Density decimal lgarithm: integers 1, 2 and 3 crrespnd t 10, 100 and 1000 secnds, respectively. The fit fr NE and PS service types is better than fr the tw ther types. Hwever, the nrmal curve seems a reasnable apprximatin fr all service types. 7

8 Part 1. Service Times lgnrmal? (7) EX Histgrams f Service Times versus Lgnrmal Density 8

9 Part 1. Service Times lgnrmal? (8) EX Service Times versus lgnrmal-quantiles A gd fit t a straight line up t 30 minutes (1800 sec) the center and the upper-right crner f bth graphs include nly a small number f large service times 9

10 Part 1. Service Times lgnrmal? (9) EXLg 10 (Service Times) versus nrmal-quantiles a mre balanced plt (main bulk f the data in the middle) the nrmal prbability plt is the mst ppular f QQ-plts a straight line in the middle f the graph is bserved with sme nise at the edges. 10

11 Part 1. Service Times lgnrmal? (10) EX Service Times versus expnential-quantiles mderately cnvex, belw the 45 line The behavir f the QQ-plt demnstrates that the sample data has a heavier tail than the theretical distributin. NOTE: QQ-plts are an excellent tl t cmpare tails f distributins; fr example, a plt can shw that the expnential tail is a gd apprximatin even if the expnential hypthesis is strngly incnsistent fr small values. 11

12 Part 1. Service Times lgnrmal? (11) EX Service Times versus nrmal-quantiles strngly cnvex, belw the 45 line the sample data has a heavier tail than the theretical distributin. 12

13 Part 1. Service Times lgnrmal? (12) Cnclusin The lgnrmal mdel prvides a gd apprximatin fr the service time distributin f the fur majr service types. The fit fr NE and PS service type is better than fr IN and NW. Why Lgnrmal? Lgnrmal distributin arises frequently in applicatins. We d nt have a gd stry behind this distributin that can explain, even partially, its prevalence. It is nt clear whether the lgnrmal distributin is s special. Apparently, ne can fit t lgnrmal data, as successsfully, als ther rich enugh families f distributins, fr example Gamma. 13

14 Part 2. Hazard Rate Functins The hazard rate functin h(t) uniquelydetermines the distributin f a nn-negative randm variable =1 ()= exp h t Π dt cntinuus time 1 h i discrete time CntinuusCase: Fr a cntinuus nn-negative randm variable T, Discrete Case: If T is a discrete nn-negative randm variables that takes values <! < with crrespnding prbabilities # $,& 1, then its hazard-sequence, fr i>0, is defined by 14

15 Part 2. Hazard Rate Functins (2) Theretical Calculatin * =1 =( )>+ =0.1( ) >+ +0.6( ) +)! >+ +0.3( ) +) 1 >+ = 15

16 Part 2. Hazard Rate Functins (3) Hw d we estimate hazard rate frm data? Asimulatin experiment: 10,000 independent realizatins f service times were simulated in Excel. The theretical hazard rates were pltted and cmpared against estimates f the hazard rate, based n the simulatin data. Cmments: The hazard-rate is neither increasing nr decreasing: hump pattern. Value at =0: 1/3*0.1 -prduct f rate f the initial phase and exit prbability. Limit at = : 1/5 -rate f the lngest final phase (exp(1/5)) 16

17 Part 2. Hazard Rate Functins (4) Estimating the Hazard Rate (34 5, red dts n pg. 12): 17

18 Part 2. Hazard Rate Functins (5) Estimating the Hazard Rate (34 5, red dts n pg. 12): -cntinued Hw d we use the estimates, 34 5? 6 =Π < 89 : = 6 ; , =0,1, If :~exp >,8[:]= A. Hence, 8 9 : = B CDCEF # HIHJKLJMK 18