Outline. Finite source queue M/M/c//K Queues with impatience (balking, reneging, jockeying, retrial) Transient behavior Advanced Queue.

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1 Outline Finite source queue M/M/c//K Queues with impatience (balking, reneging, jockeying, retrial) Transient behavior Advanced Queue Batch queue Bulk input queue M [X] /M/1 Bulk service queue M/M [Y] /1 Erlangian queue Erlangian queue M/E k /1 Erlangian queue E k /M/1 1

2 Queues with finite source, M/M/c//K queue Infinite calling population Arrival rate is constant, λ Example, the outpatient of hospital, passengers of bus station, etc. Finite calling population Arrival rate is variant Example, K machines, breakdown rate λ, service rate of maintenance is μ, # in maintenance is n, arrival rate is (K-n)λ 2

3 M/M/c//K Population size: K Arrival rate of outside customers: (K-n)λ Service rate of each server: μ Special case of birth-death process Birth rate: Death rate: n ( K n) n ( nc) 3

4 State transition rate diagram State transition rate diagram K ( K 1) ( K 2) ( Kc1) ( K c) ( K c1) c c+1 K 2 3 c c c With local balance equation, we have n K!/ ( K n)! r n 0, n 0,1,..., c 1 n! K!/ ( K n )! r n 0, n c, c 1,..., K nc c c! c 4

5 Steady state distribution Steady state distribution of M/M/c//K n K n r 0, n 0,1,..., c 1 n K n! n r 0, n c, c 1,..., K nc n c c! π 0 can be calculated with the Law of total probability, but the form is complicated 5

6 Special case: M/M/1//K When c=1, we have n K K! 0 ( )! n n n! 0 n K n and 0 K n0 n K! ( K n)! 1 Analysis of other metrics is similar 6

7 Average queue length and mean waiting time Average number of customers in the system L K n n n0 Effective arrival rate K ( K n) ( K L) eff n0 Average queue length eff Lq L L ( K L) r n Mean waiting time and sojourn time W L q eff Lq L L T ( K L) ( K L) eff 7

8 Example: models with spares M/M/c//K queue can model the production system with spares (Gross book p.88) n: number of failed machines c: number of technicians to repair machine At most K machines in operation, Y spares Every machine has a failure rate λ; when fails, if has spares, immediately replace the failed one; when repaired, become spare Analyze the queue similar to M/M/c//K When Y, equivalent to M/M/c queue 8

9 Queues with impatience Customer is not patient during its waiting period model for practical situation Call center, crowded service facility, etc. types of queues with impatience Queues with balking (not joining when crowded) Queues with reneging (leave the line with probability) Queues with jockeying (moving among multiple lines) Retrial queues (balk when crowded and rejoin the line after a random time) 9

10 M/M/1 with Balking One example is finite storage queue M/M/c/K Balking function is to decrease the arrival rate when system becomes crowded (discouraged) {b n }, b n+1 < b n for all n=0,1,2, arrival rate: λ n =λb n for example, b n =1/n, or 1/n 2, or e -αn Waiting time is more reasonable than queue length, so Estimate the mean service time and set b n = e -αn/μ 10

11 M/M/1 with reneging All arrivals join the queue, but estimate the waiting time and leave the line if with long waiting time Reneging function r(n), total leaving rate at n r(0)=r(1)=0 Equivalent to a variant service rates, μ+r(n) Similar analysis to birth-death process Example: r(n) = e αn/μ, n>1 11

12 Other queues with impatience Queues with jockeying Difficult to analyze, no meaningful results Retrial queues Intensively studied, good model for many practical systems, such as call center Particular results with insights to queuing theory Discuss it later 12

13 Transient behavior Balance equation for equilibrium behavior or steady-state distribution Analysis for transient behavior is much more complicated M/M/1/1 and M/M/1, as example 13

14 Transient behavior of M/M/1/1 Differential-difference equation (optional) So, 0 1 We have d 0() t 1( t) 0( t) dt d1() t 1( t) 0( t) dt ( t) ( t) 1 ( t) ( ) ( t) 1 1 0( t) [ 0(0) ] e 1( t) [ 1(0) ] e Chalk drawing, curve of probability ( ) t ( ) t 14

15 Transient behavior of M/M/1 Differential-difference equation (optional) ( t) ( ) ( t) ( t) ( t), n 0 n n n1 n1 ( t) ( t) ( t) Z-transform and Laplace-transform Sum all of the equations and have P( z, s) i z 1 (1 z) P0 ( s) ( s) z z With initial condition n(0)=i 2 15

16 Transient behavior of M/M/1 Reverse transform and obtain (optional) p t e I t I t ( ) t ( ni)/2 ( ni1)/2 n( ) ni (2 ) ni1(2 ) (1 n j/2 ) I j (2 t ) jni2 where modified Bessel function of the first kind n2k ( y / 2) In( x) :, n 1 k!( n k)! k0 16

17 Advanced Markovian queues Batch queue Bulk input queue M [X] /M/1 Bulk service queue M/M [Y] /1 Erlangian queue Erlangian queue M/E k /1 Erlangian queue E k /M/1 17

18 Bulk input queue M [X] /M/1 Batch arrival, Poisson process, arrival rate λ number of customers in each arrival is an integer random number X Probability of X is c n =Pr[X=n] State transition rate diagram X={1,2} as example, pp.118 of Gross book 18

19 Balance equation of M [X] /M/1 Global balance equation 0 ( ) p p p c, n 1 0 p p 0 1 n n1 nk k k 1 n To solve these equations, use Z-transforms of c and p 19

20 Balance equation of M [X] /M/1 Multiply z n on both sides and sum all 0 We have Thus, n n n n n pnz pnz pnz pnkck z n0 n1 z n1 n1 k 1 0 P( z) [ P( z) p0] [ P( z) p0] C( z) P( z) z p (1 ) 0 z P( z), z 1 (1 z) z[1 C( z)] 20

21 Average metrics of M [X] /M/1 Let z=1, obtain the average metrics p 1, r /, re[ X ] 0 L r E X E X re X 2(1 ) 2(1 ) 2 2 ( [ ] [ ]) [ ] L L (1 p ) L q T L /, W L / 0 q q 21

22 Example of M [X] /M/1 X is a constant K, apply the results we have L K K 1, K 2(1 ) 2 1 What s the difference from M/M/1 with Kλ? Equivalent to M/M/1 with Kλ and a scale factor (K+1)/2 Worse performance metrics Why? more bursty 22

23 Bulk service queue M/M [Y] /1 Similar to M/M/1, except the server serves K customers at a time, called M/M [K] /1. K is a constant. Two model for different service mode Partial-batch Start service no matter n<k Full-batch Start service until n>=k 23

24 M/M [K] /1 v.s. M/M/1 formulas of metrics are similar Formulas for distribution, L, T, L q, W q are similar similar to M/M/1 with service rate Kμ r 0 is the root of characteristic equation ρ=λ/kμ is the utilization factor of M/M/1 which one is bigger? Guess: r 0 > ρ (because M/M [K] /1 is more bursty more crowded) Metrics of M/M [K] /1 is worse than those of M/M/1 24

25 Erlangian queues and M/E k /1 Exponential distribution Memoryless property, simplify the analysis Limitation, more general cases Erlang distribution for service time and interarrival time Sum of K-stage exponential distributions More general model Other distributions PH, Coxian, hyperexponential, etc. 25

26 Erlang distribution k-stage Erlang distribution, each stage is exponentially distributed with rate kμ By convolution, pdf is f ( x) E[ X ], Var[ X ], c 2 X k k k( kx) ( k 1)! k1 e kx kμ kμ kμ kμ 1 2 i k 26

27 Erlang distribution Distribution with 2-parameters, c x <1, to model smooth stochastic process Approximate empirical distribution, use the mean and variance to match μ and k If c x >1, we can use other distribution model, such as hyperexponential distribution 27

28 M/E k /1 queue Service time is Erlang-k distribution k stage, each stage is exponential with kμ State definition (n,i): n customers and customer in service is in phase i, i=1,2,,k. (k is the first phase and 1 is the last phase) Global balance equation 28

29 Average metrics of M/E k /1 Consider equivalent queue with state (n-1)k+i, # of phase requests Equivalent to M [k] /M/1, average # of customer k 1 is average # of phase requests L, So, 2 k1 k1 Wq, Lq Wq 2 k (1 ) 2 k (1 ) T Wq, L Lq Can we use PASTA to do mean value analysis? 29

30 Example of M/E k /1 Suppose a queuing system λ=16/h, Poisson arrival Mean service time 2.5min, std. deviation is 1.25min not exponentially distribution Use Erlang-k to approximate, C X =0.5=(1/k) 0.5, k=4 M/E 4 /1 queue, with λ=4/15/min, μ=0.4/min, ρ=2/3 So, L q =5/6, W q =25/8min 30

31 E k /M/1 queue The arrival process is not Poisson arrival Erlang-k distribution with mean 1/λ, each stage is exponential distribution with kλ State: (n,i), or change to nk+i The balance equation is identical to that of full-batch bulk-service queue M/M [k] /1 with kλ arrival and constant batch size k 31

32 Average metrics of E k /M/1 The distribution is similarly as p (1 r ) r, j k 1, / ( P) j The distribution of E k /M/1 is We have k 0 jk 0 0 nk k 1 p p (1 r ) r (1 r... r ) n jnk (1 r ) r ( P) nk k k 1 j k nk k 0 0 L, Lq L, T L /, Wq T 1 r 1 32

33 Midterm Exam Midterm Exam 1.5 hours 33