Monotonicity, Convexity and Comparability of Some Functions - PowerPoint PPT Presentation

Monotonicity, Convexity and Comparability of Some Functions Associated with Block-Monotone Markov Chains and Their Applications to Queueing Systems Hai-Bo Yu College of Economics and Management Beijing University of Technology, Beijing, China MAM9, Budapest, Hungary June 28-30, 2016

Outline 1. Introduction 2. Block-Monotonicity of Probability Vectors and Stochastic Matrices 3. Block-Monotone Markov Chains 4. Application to the GI/Geo/1 Queue 5. Numerical Examples 6. Conclusion 2

1. Introduction Queueing Systems & Univariate Markov chains  – Research in this area started in 1950’s – Classical queues: M/M/1 , M/G/ 1, GI/M/1, GI/Geo/ 1,… Imbeeded Markov chain: (e.g., Kendall 1951,1953),  Hunter (1983), Tian and Zhang (2002)) Stochastic comparison methods: (e.g.,Stoyan (1983))  Monotonicity and convexity of functions w.r.t.  univariate Markov chains (Yu, He and Zhang (2006)) Question : Whether above results may be extended to  bivariate Markov chains? 3

1. Introduction (continued) Bivariate Markov chains with block-structured  transition matrices (e.g., QBD) – Research in this area started in 1980’s. – Continuous-time queues: GI/PH/1 , PH/G/1,MAP/PH/1, … – Discrete-time queues: GI/Geo/1 , GI/G/1, … – Methods: Matrix-analytic method (MAM) vs. block-monotone Markov chain MAM: (Neuts (1981,1989), Gibson and Seneta (1987),  Zhao, Li and Alfa (2000); Tweedie(1998), Liu (2010), He(2014), Alfa (2016), etc.) 4

1. Introduction (continued) Block-monotone Markov chain approach  – First introduced the stochastic vectors based on F-orderings (Li and Shaked (1994)) – Defined the block-increasing order, and proved the stationary distributions of its truncations converge to that of the original Markov chain with monotone transition matrix (Li and Zhao (2000)) – Provided error bounds for augmented truncations of discrete-time block-monotone Markov chains under geometric drift conditions (Masuyama (2015)). – Continuous-time block-monotone Markov chain(Masuyama (2016) ). 5

2. Block-Monotonicity of Probability Vectors and Stochastic Matrices Definition:Block Stochastic orders  – a =( a 0 , a 1 , a 2 , …), b =( b 0 , b 1 , b 2 , …), where a n =( a n (1), a n (2) , …, a n (m) ), b n =( b n (1), b n (2) , …, b n (m) ). ≤ ≤ a b , if a E s b , for s = 1, 2 E s − − m s st m el m where       I O O O I O O O m m       2 I I O O I I O O     m m m m   =   , (1) = 2  E 3 I 2 I I O  E I I I O m  m m m  m  m m m    4 I 3 I 2 I I    I I I I  m m m m m m m m                   ≤ Note: For s = 1, Li and Zhao (2000) defined the order − − m 1 st 6

2. Block-Monotonicity of Probability Vectors and Stochastic Matrices (continued) Example 1. 1) a 0 =(0.3 , 0.2), a 1 =(0.3 , 0.2), ≤ b 0 =(0.2 , 0.2), b 1 =(0.4 , 0.2), then a b . − − m 1 st ≤ 2) a =( a 0 , a 1 , a 2 , …), b =( a 0 + a 1 , a 2 , a 3 , …), then a b − − m 1 st Example 2. a 0 =(0.25 , 0.25), a 1 =(0.15 , 0.15), a 2 =(0.1 , 0.1), b 0 =(0.21 , 0.21), b 1 =(0.17 , 0.17), b 3 =(0.12 , 0.12), ≤ then a b . − − m 2 st 7

2. Block-Monotonicity of Probability Vectors and Stochastic Matrices (continued) Definition: Block-Monotone Stochastic Matrices  Let P = ( A k,l ), A k,l = ( A k,l ( i,j )), i,j=1,2, … ,m k,l=0,1,2, … − ≥ ∈ 1 s E PE O , if for s = 1, 2 . P M − − m m el m s st ∈ P M – For n = 1, Definition of (Masuyama (2015)) − − m 1 st ≤ ⇔ ∈ P M – For n = 1, a b implies − − − − m 1 st m 1 st ≤ a P b P (Li and Zhao (2000)) − − m 1 st 8

2. Block-Monotonicity of Probability Vectors and Stochastic Matrices (continued) Example 3 :QBD  ≤ ≤ + ≤ + + i)If , , and , B C F A A F B F A 1 0 el el 2 el    B C O O   ∈ then P M  F A A O   − − 1 m st 1 0   =  P O A A A ≤ + ≤ + +  2 1 0  F B A F 2 A A A ii)If , ,and el 0 el 2 1 0  O O A A   2 1   + ≤ + + ∈ B 2 C F 2 A 3 A      P M   ,then − − 1 0 el m 2 st Example 4. QBD for GI/Geo/1 queue (Alfa 2016)        0 0 0 0 g g g g 1 2 3 m     = = µ   F A B 1 0 0 0 0 0 0 0     2     = =   = µ + µ B 0 1 0 0 C 0 0 0 0 A B C     1               = µ     A C       0 0 1 0 0 0 0 0 0 ∈ ∈ Then , . P M P M − − − − m 1 st m 2 st 9

2. Block-Monotonicity of Probability Vectors and Stochastic Matrices (continued) ∈ Property 1: Suppose that , s =1,2, then  P M − − m s st ⇒ ≤ i. a b ≤ for n =1,2, ….. n n aP bP − − m s st − − m s st + ≤ ⇒ ≤ n n 1 aP aP ii. for n =1,2, …. a aP − − − − m s st m s st ⇒ ≤ + ≤ n 1 n aP a aP aP − − − − for n =1,2, …. m s st m s st 10

3. Block-Monotone Markov Chains (continued) n ∈ N Theorem 1: For a DTMC Z = { Z n , } with transition  0 v matrix P having block size m and initial distribution . = < ∞ n T Assume that E [ ( f Z )] v P f v n 1) If Condition I m-s-st , II m-s-st and III m-s-st in Eq.(2) hold, then is increasing concave in n for n=0,1,2,… E [ ( f Z ) ] v n and s=1 ,2. 2) If Condition I m-s-st , II’ m-s-st and III m-s-st in Eq.(2) hold , then is decreasing convex in n for n=0,1,2,… E [ ( f Z ) ] v n and s=1 ,2. 11

3. Block-Monotone Markov Chains (continued) For s = 1, 2, ∈ Condition I m-s-st : P M − − m s st ≤ Condition II m-s-st : v v P − − m s st ≤ Condition II’ m-s-st : (2) v P v − − m s st ≤ h T Condition III m-1-st : is decreasing w.r.t . orders − − m s st f where = − T T T h Pf f f 12

3. Block-Monotone Markov Chains (continued) { } n ∈ N Theorem 2: For two DTMCs and  Z= , Z n 0 { }   n ∈ N Z= , Z with the same state space and initial n 0  v distribution , If their transition matrices and satisfy: P P   ∈ ≤ ∈ P M P P , and either or , then P M − − − − − − m s st m s st m s st  ≤ E E for n=0,1 ,2,…, and s=1 ,2. [ ( f Z )] [ ( f Z ) ] v n v n Proof based on Property 2:    ≤ ≤ s s Property 2: Suppose (i.e., ) P P PE PE  − − − − m m s st m m s st ∈  ≤ ∈ n n and either or , then P M P P P M − − − − m s st − − m s st m s st Note : Li and Zhao (2000) gave result in property 2 for s =1 13

4. Application to the GI/Geo/1 Queue There is a single server. Service times follow the  µ < µ < geometric distribution with parameter , 0 1 Service discipline: first-come-first-served (FCFS).  Inter-arrival times are follow a general  distributions , g =( g 1 , g 2 , …, g m ), m< ∞ , and with DPH representation ( β , B) of order m, where β =( g 1 , g 2 , …, g m ), B is given in Eq. (4). 14

4. Application to the GI/Geo/1 Queue (continued) DTMC Z = {( , ) , n=0,1,2,… }  I J n n I : the number of customers in the system at time n  n J : the remaining inter-arrival time at time n  n The transition matrix P given by     B C O O    A A A O   2 1 0   =  P O A A A   2 1 0 (3)    O O A A 2 1          15

4. Application to the GI/Geo/1 Queue (continued)    0 0 0 0   where  1 0 0 0     (4) =  B 0 1 0 0               0 0 1 0    g g g g 1 2 3 m    0 0 0 0   (5)   =  C 0 0 0 0               0 0 0 0 = µ = µ + µ = µ A B A B C A C , , (6) 0 2 1 16

4. Application to the GI/Geo/1 Queue (continued) For the GI/Geo/1 Queue Corollary 1:  1)If Condition I m-1-st in Eq.(7), II m-1-st in Eq.(8) and E [ ( f Z ) ] II m-1-st in Eq.(9) hold, then is increasing v n concave in n . 2)If Conditions I m-1-st in Eq.(7), II’ m-1-st in Eq.(8’) E and II m-1-st in Eq.(9) hold, then is [ ( f Z ) ] v n decreasing convex in n. 17