on truncations for a class of finite markovian queuing models

ON TRUNCATIONS FOR A CLASS OF FINITE MARKOVIAN QUEUING MODELS Yacov Satin Vologda State University S.Orlova, 6, Vologda, Russia

Alexander Zeifman Anna Korotysheva Vologda State University, Ksenia Kiseleva Vologda, Russia Vologda State University Institute of Informatics Problems, S.Orlova, 6, Vologda, Russia FRC CSC RAS; ISEDT, RAS Victor Korolev Moscow State University, Leninskie Gory, Moscow, Russia Institute of Informatics Problems, FRC CSC RAS

KEYWORDS Markovian queueing models; SZK model; approximation bounds ABSTRACT We consider a class of finite Markovian queueing models and obtain uniform approximation bounds of truncations.

{0, 1, . . . , r} with possible batch arrivals and group services. Let X(t), t ≥ 0 be the queue-length process for the queue, pij (s, t) = P {X(t) = j |X(s) = i }, i, j ≥ 0, 0 ≤ s ≤ t, be transition probabilities for X = X(t), and pi (t) = P {X(t) = i} be its state probabilities. Throughout the paper we assume that (

INTRODUCTION

=

P (X (t + h) = j|X (t) = i) = qij (t) h + αij (t, h) , if j 6= i, P 1− qik (t) h + αi (t, h) , if j = i,

(1)

k6=i

It is well known that explicit expressions for the probability characteristics of stochastic models can be found only in a few special cases, moreover, if we deal with an inhomogeneous Markovian model, then we must approximately calculate the limiting probability characteristics of the process. The problem of calculation of the limiting characteristics for inhomogeneous birth-death process via truncations was firstly mentioned in (Zeifman 1991) and was considered in details in (Zeifman et al. 2006). In (Zeifman et al. 2014b) we have proved uniform (in time) error bounds of truncation this class of Markov chains. First uniform bounds of truncations for the class of Markovian time-inhomogeneous queueing models with batch arrivals and group services (SZK models) introduced and studied in our recent papers (Satin et al. 2013, Zeifman et al. 2014a), were obtained in (Zeifman et al. 2014c). In this note we deal with approximations of finite SZK model via the same models with smaller state space and obtain the correspondent bounds of error of truncation bounds.

where all αi (t, h) are o(h) uniformly in i, i. e., supi |αi (t, h)| = o(h). We also assume qi,i+k (t) = λk (t), qi,i−k (t) = µk (t) for any k > 0. In other words, we suppose that the arrival rates λk (t) and the service rates µk (t) do not depend on the queue length. In addition, we assume that λk+1 (t) ≤ λk (t) and µk+1 (t) ≤ µk (t) for any k and almost all t ≥ 0. Hence, X(t) is a so-called SZK model, which was studied in (Satin et al. 2013, Zeifman et al. 2014a, 2014c). We suppose that all intensity functions are locally integrable on [0, ∞), and

Consider a time-inhomogeneous continuous-time Markovian queueing model on the state space E =

dp = A(t)p(t), dt

Proceedings 29th European Conference on Modelling and Simulation ©ECMS Valeri M. Mladenov, Petia Georgieva, Grisha Spasov, Galidiya Petrova (Editors) ISBN: 978-0-9932440-0-1 / ISBN: 978-0-9932440-1-8 (CD)

λk (t) ≤ λk ,

µk (t) ≤ µk ,

(2)

for any k and almost all t ≥ 0, and put Lλ =

r X i=1

λi ,

Lµ =

r X

µi .

(3)

i=1

Then the probabilistic dynamics of the process is represented by the forward Kolmogorov system (4)

where A(t) = µ1 (t)

µ2 (t)

···

a11 (t)

µ1 (t)

···

λ1 (t)

a22 (t)

···

λr−1 (t)

λr−2 (t)

···

arr (t)

µk (t) −

Pr−i

λk (t).

 a (t) 00  λ1 (t)   =  λ2 (t)   ··· λr (t)

and aii (t) = −

Pi

k=1

µr (t)  µr−1 (t)    µr−2 (t)  ,  

k=1

(5)

Similarly, instead of (8), we obtain the corresponding reduced system for the truncated process in the form: dzN −1 = BN −1 (t)zN −1 (t) + fN −1 (t), (10) dt >

where fN −1 (t) = (λ1 , · · · , λN −1 ) , zN −1 (t) = > N −1 (p1 , p2 , · · · , pN −1 ) , BN −1 = (bij ∗ (t))i,j=1 =

Throughout the P paper by k · k we denoteP the l1 norm, i. e., kxk = |xi |, and kBk = supj i |bij | for B = (bij ). Let Ω be the set all stochastic vectors, i. e., l1 vectors with nonnegative Pr coordinates and unit norm. Hence kA(t)k ≤ 2 k=1 (λk (t) + µk (t)) ≤ 2 (Lλ + Lµ ) for almost all t ≥ 0.

      

TRUNCATIONS Consider the family of “truncated” processes XN −1 (t), and let EN −1 = {0, 1, . . . , N − 1} be the corresponding state space and AN −1 = µ2

···



b00

µ1

   =  

µ3

µN −1



λ1

b11

µ1

µ2

λ2

λ1

b22

µ1

···

µN −2

···

µN −3

    (6)  

λN −2

λN −3

···

λ1

bN −1,N −1

··· λN −1

Instead of (4), for XN −1 (t) we have the following forward Kolmogorov system: dpN −1 = AN −1 (t)pN −1 . dt Setting p0 (t) = 1− the equation

(7)

P

i≥1 pi (t), from (4) we obtain

dz = B(t)z(t) + f (t), dt

(8) >

where f (t) = (λ1 , λ2 , · · · , λ
r ) , > (p1 , p2 , · · · , pr ) , B = bij (t)ri,j=1 =       

a11 − λ1

µ1 − λ1

···

λ1 − λ2

a22 − λ2

···

λ2 − λ3

λ1 − λ3

···

λr−2 − λr

···

··· λr−1 − λr

z(t)

µr−1 − λ1  µr−2 − λ2    µr−3 − λ3  .   arr − λr

=

···

µN −1 − λ1

λ1 − λ2

b22 − λ2

···

µN −2 − λ2

λ2 − λ3

λ1 − λ3

···

µN −3 − λ3

λN −3 − λN −1

···

bN −1,N −1 − λN −1 (11)

···

dzN −1 = BN −1 (t)zN −1 (t) + f (t), dt

   .  

(12)

One can see that the solution of system (10) and the corresponding solution of system (12) with initial condition p0 (0) = 1 coincide. Below we will identify the vector with entries, say, (a1 , . . . , aN −1 )> and the r-dimensional vector with the same first N − 1 coordinates and the rest equal to zero. Let {di }, i = 1, 2, . . . be an increasing sequence of positive numbers, d1 = 1, and Pi i X dk dn . (13) W = min k=1 , gi = i≥1 i n=1 Put

k≥1

αi (t) = −aii (t) + λr−i+1 (t) − (14) i−1 dk+1 X dk (λk (t) − λr−i+1 (t)) − (µi−k (t) − µi (t)) , di di k=1

and α(t) = min αi (t).

(15)

i≥1

Let D be upper triangular matrix,  d1 d1 d1 · · ·  0 d2 d2 · · ·   0 d3 · · · D= 0  · · · · ·· ··· ···  0 0 0 ···

d1 d2 d3 ··· dr

    ,  

(16)

and let k • k1D be the corresponding norm kzk1D = kDzk1 . Then the important inequality Z t kV (t, s)k ≤ exp γ(B(u)) du s

(9)



Consider the system

r−i X

beP the corresponding intensity matrix, where bii (t) = PN −1−i i − k=1 µk (t) − k=1 λk (t).

µ1 − λ1

λN −2 − λN −1

By E(t, k) = E {X(t) |X(0) = k } denote the mathematical expectation of the process at a moment t under the initial condition X(0) = k. Recall that a Markov chain X(t) is called weakly ergodic, if kp∗ (t) − p∗∗ (t)k → 0 as t → ∞ for any initial conditions p∗ (0), p∗∗ (0), where p∗ (t) and p∗∗ (t) are the corresponding solutions of (4), and a Markov chain X(t) has the limiting mean ϕ(t), if limt→∞ (ϕ(t) − E(t, k)) = 0 for any k.

b11 − λ1

holds, where V (t, s) = V (t)V −1 (s) is the Cauchy matrix of equation (8), and γ(B(t)) is the logarithmic norm of the matrix function B(t), see details in (Van Doorn et al. 2010, Granovsky and Zeifman 2004,

Zeifman et al. 2008). Further, for an operator function from l1 to itself we have the simple formula   X γ(B(t)) = sup bjj (t) + |bij (t)| . j

On the other hand, ˆ B(t)z N −1 (t) = (B(t) − BN −1 (t)) zN −1 (t) = ((a11 (t) − b11 (t))pN −1,1 (t), · · · , >

(aN −1,N −1 (t) − bN −1,N −1 (t))pN −1,N −1 (t))

i6=j

Hence we obtain the following bound for the logarithmic norm of B(t):

and ˆ kB(t)z N −1 (t)k1D = kD (B(t) − BN −1 (t)) zN −1 (t)k1 = d1 (b11 (t) − a11 (t))pN −1,1 (t) + (d1 + d2 )(b22 (t) − a22 (t))pN −1,2 (t) + · · · +

γ(B(t))1D = γ(DB(t)D−1 ) = sup{−αi (t)} = −α(t),

(17)

i≥1

(d1 + · · · + dN −1 )(bN −1,N −1 (t) − aN −1,N −1 (t))pN −1,N −1 (t) = X X λk (t)pN −1,1 (t) + (d1 + d2 ) λk (t)pN −1,2 (t) + · · · +

d1

k≥N −1

k≥N −2

(d1 + · · · + dN −1 )

where DBD 

(26) ,

a11 − λr     d  (λ1 − λr ) 2 d1     ···       (λr−1 − λr ) dr d1

−1

=

(18)

d (µ1 −µ2 ) 1 d2

···

d (µr−1 −µr ) 1 dr

a22 − λr−1

···

d (µr−2 −µr ) 2 dr

···

···

···

d (λr−2 − λr−1 ) r d2

···

arr − λ1

       .       

ˆ Now we will estimate kB(t)z N −1 (t)k1D . Firstly, kzN −1 (t)k1D∗ ≤ kV (t)k1D∗ kzN −1 (0)k1D∗ + Z t kV (t, τ )k1D∗ kfN −1 (τ )k1D∗ dτ ≤ (28) 0 ∗

kV (t, s)k1D ≤ e−

s

α(u) du

M ∗ e−a t kzN −1 (0)k1D∗ + .

(19)

Now let {di } and {d∗i } be two increasing sequences such that d1 = d∗1 = 1, all di < d∗i , i ≥ 2, and the following assumptions hold: kV (t, s)k1D ≤ M e−a(t−s) and

∗

kV (t, s)k1D∗ ≤ M ∗ e−a

(t−s)

∗ KN M∗ , a∗

∗ because kfN −1 (t)k1D∗ ≤ KN for almost all t ≥ 0.

(20)

Put X(0) = XN −1 (0) = 0, then zN −1 (0) = 0, hence K∗ M ∗ (29) kzN −1 (t)k1D∗ ≤ N ∗ , a

(21)

for any t ≥ 0.

for any 0 ≤ s ≤ t, and some positive numbers M, M ∗ , a, a∗ .

For definiteness suppose that N is odd. All pN −1,i (t) ≥ 0, then

∗ be a positive number such that Let KN ∗ d∗1 λ1 +(d∗1 +d∗2 )λ2 +· · ·+(d∗1 +· · ·+d∗N −1 )λN −1 ≤ KN . (22)

For bounding the truncation error we rewrite (12) as dzN −1 ˆ = B(t)zN −1 (t) + f (t) − B(t)z N −1 (t), (23) dt ˆ = B(t) − BN −1 (t). Then we have where B(t) Z t zN −1 (t) = V (t)zN −1 (0) + V (t, τ )f (τ ) dτ − 0 Z t ˆ )zN −1 (τ ) dτ. (24) V (t, τ )B(τ 0

kzN −1 (t)k1D∗ =

X

pN −1,i (t)

i≥1

X

d∗k ≥

k=1 N −1 X

d∗i pN −1,i (t) ≥

i≥ N 2−1

i X

d∗i pN −1,i (t).

(30)

i= N 2−1

On the other hand we have the bound: X

d1

λk (t)pN −1,1 (t) +

k≥N −1

(d1 + d2 )

X

λk (t)pN −1,2 (t) + · · · +

k≥N −2

(d1 + · · · + dN −1 )

X

λk (t)pN −1,N −1 (t) ≤

k≥1

N −1 Hence, if z(0) = zN −1 (0) = 0, then the sum of the 2 X X (d1 + · · · + d N −1 ) λk (t) pN −1,k (t) + first and the second summands gives us z(t), and we 2 k=1 k≥ N −1 obtain in any norm the bound 2  X Z t λk (t) (d1 + · · · + d N −1 )pN −1, N −1 (t) + · · · + 2 2 ˆ )zN −1 (τ ) dτ k ≤ k≥1 kz(t) − zN −1 (t)k ≤ k V (t, τ )B(τ 0 (d1 + · · · + dN −1 )pN −1,N −1 (t)) . Z t ˆ )zN −1 (τ )k dτ. (25) kV (t, τ )kkB(τ

0

λk (t)pN −1,N −1 (t).



Therefore Rt

X k≥1

(31)

(27)

∗ Moreover, we have KN = 2, d∗N −1 = 2 2 Theorem 1 yields the bounds

P Denote ΛK = k≥K λk , where λk are defined by (2). Then from (3), (27), (30) and (31) we obtain ˆ kB(t)z N −1 (t)k1D ≤ g N −1 Λ N −1 2

 Lλ

2

N −1 2 X

kp(t) − pN −1 (t)k ≤

pN −1,k (t) +

k=1

 g N −1 pN −1, N −1 (t) + · · · + gN −1 pN −1,N −1 (t) ≤ 2 2  gN −1 ∗ g N −1 Λ N −1 + Lλ ∗ d N −1 pN −1, N −1 (t)+ d 2 2 2 2 N −1

(32)


∗ · · · + dN −1 pN −1,N −1 (t) ≤ g N −1 Λ N −1 + 2

2

∗ gN −1 KN gN −1 M∗ kzN −1 (t)k1D∗ ≤ g N −1 Λ N −1 + Lλ ∗ , ∗ ∗ d N −1 d N −1 a 2 2 2

N −1 2

,

, and

(36)

and

2

Lλ

4N 9·2

N −1 2

4N

|E(t, 0) − EN −1 (t, 0)| ≤

N −1 , 9·2 2 for any t ≥ 0, where X(0) = XN −1 (0) = 0.

(37)

Hence, if N = 41, then the truncation error for vector of state probabilities and for the mathematical expectation of the process X(t) smaller than 2 · 10−5 .

2

for any t ≥ 0. Finally, from (32) we obtain the following bound of truncation error: kz(t) − zN −1 (t)k ≤ 

 t

Z

Me 0

−a(t−τ )

∗ gN −1 KN M∗   g N −1 Λ N −1 + Lλ ∗  dτ ≤ d N −1 a∗ 2 2

(33)

2





∗ M  gN −1 KN M∗  g N −1 Λ N −1 + Lλ ∗ . a d N −1 a∗ 2 2 2

P Now consider k • k1E norm: kzk1E = n|pn |, then kzk1E ≤ W −1 kzk1D , see, for instance, (Zeifman et al 2006), and we obtain the following statement.

Fig. 1. [0, 10].

First example, approximation of the mean E(t, 0) on

Theorem 1. Let (20) and (21) hold. Then X(t) is exponentially weakly ergodic, has the limiting mean, say, E(t, 0), and the following bounds of truncation error hold: kp(t) − pN −1 (t)k ≤ ∗ M M∗  gN −1 KN g N −1 Λ N −1 + Lλ ∗ , 2 2 a d N −1 a∗

(34)

2

and |E(t, 0) − EN −1 (t, 0)| ≤ ∗ M  M∗  gN −1 KN g N −1 Λ N −1 + Lλ ∗ , 2 2 aW d N −1 a∗

Fig. 2. First example, approximation of the probability of empty queue P{X(t) = 0|X(0) = 0} on [0, 10].

(35)

2

for any t ≥ 0, where X(0) = XN −1 (0) = 0, and EN −1 (t, k) = E {XN −1 (t) |XN −1 (0) = k } is the mathematical expectation of the truncated process at the moment t under initial condition XN −1 (0) = k. EXAMPLES Let r = 1010 , λ∗ (t)∗ = 1 + sin(2πt), µ∗ (t) = 4 + cos(2πt), µi (t) = µ i(t) .

2. Let λi (t) =

Put all dk = 1. Then we have α(t) = µ∗ (t), Lλ = 1, gN −1 ≤ N , W = 1, and (20) holds for M = 1, a = 3. Let now d∗k+1 = 2k . Then the respective α∗ (t) = µ∗ (t) − λ∗ (t), and (21) holds for M ∗ = 1, a∗ = 1. N −1 ∗ Moreover, we have KN = 2, d∗N −1 = 2 2 , and 2 Theorem 1 yields the bounds

1. Let λ1 (t) = λ∗ (t), λi (t) = 0, for i ≥ 2. Put all dk = 1. Then we have α(t) = µ∗ (t), Lλ = 2, gN −1 ≤ N , W = 1, and (20) holds for M = 1, a = 3. Let now d∗k+1 = 2k . Then the respective α∗ (t) = ∗ µ (t) − λ∗ (t), and (21) holds for M ∗ = 1, a∗ = 1.

λ∗ (t) 3i .

kp(t) − pN −1 (t)k ≤

4N 9·2

N −1 2

,

(38)

and |E(t, 0) − EN −1 (t, 0)| ≤

4N

N −1 , 9·2 2 for any t ≥ 0, where X(0) = XN −1 (0) = 0.

(39)

Hence, if N = 41, then the truncation error for vector of state probabilities and for the mathematical expectation of the process X(t) smaller than 2 · 10−5 .

Zeifman, A. I. 1995b. ”On the estimation of probabilities for birth and death processes.” J. App. Probab. 32, 623–634. Zeifman, A.; S. Leorato; E. Orsingher; Ya. Satin and G. Shilova. 2006. ”Some universal limits for nonhomogeneous birth and death processes.” Queueing Syst. 52, 139–151. Zeifman, A. I.; V.E. Bening and I.A. Sokolov. 2008. Continuous-time Markov chains and models. Elex-KM, Moscow (in Russian). Zeifman, A.; V. Korolev; A. Korotysheva, Y. Satin and V.Bening. 2014a. ”Perturbation bounds and truncations for a class of Markovian queues.” Queueing Syst. 76, 205–221. Zeifman, A.I.; Y. Satin; V. Korolev and S. Shorgin. 2014b. ”On truncations for weakly ergodic non-stationary birth and death processes.” Int. J. Appl. Math. Comp. Sci.

Fig. 3. Second example, approximation of the mean E(t, 0) on [0, 10].

A. I. Zeifman, Ya. Satin, G. Shilova, V. Korolev, V. Bening, S. Shorgin. 2014c. On truncations for SZK model // Proceedings 28th European Conference on Modeling and Simulation, ECMS 2014, Brescia, Italy. 577–582.

AUTHOR BIOGRAPHIES YACOV SATIN is Candidate of Science (PhD) in physics and mathematics, associate professor, Vologda State University. His email is [email protected].

Fig. 4. Second example, approximation of the probability of empty queue P{X(t) = 0|X(0) = 0} on [0, 10].

Acknowledgement. This work was supported by the Russian Foundation for Basic Research, projects no. 13-07-00223, 14-07-00041, 15-01-01698; and by Ministry of Education and Science, State Contract No. 1816.

ALEXANDER ZEIFMAN Doctor of Science in physics and mathematics; professor, Heard of Department of Applied Mathematics, Vologda State University; senior scientist, Institute of Informatics Problems, Federal Research Center ”Computer Science and Control” of the Russian Academy of Sciences; principal scientist, Institute of Socio-Economic Development of Territories, Russian Academy of Sciences. His email is [email protected] and his personal webpage at http://uni-vologda.ac.ru/ zai/eng.html.

REFERENCES Daleckij, Ju.L. and M.G. Krein. 1974. Stability of solutions of differential equations in Banach space. Amer. Math. Soc. Transl. 43. Van Doorn, E.A., A.I. Zeifman and T.L. Panfilova. 2010. ”Bounds and asymptotics for the rate of convergence of birth-death processes.” Theory Probab. Appl. 54, 97–113. Granovsky, B.L. and A.I. Zeifman. 2004. ”Nonstationary Queues: Estimation of the Rate of Convergence.” Queueing Syst. 46, 363–388. Satin, Ya. A.; A. I. Zeifman and A. V. Korotysheva. 2013. ”On the rate of convergence and truncations for a class of Markovian queueing systems.” Theory. Prob. Appl. 57, 529– 539. Zeifman, A. I. 1991. ”Qualitative properties of inhomogeneous birth and death processes.” J. Math. Sci. 57, 32173224 (The Russian original paper was published in 1988). Zeifman, A. I. 1995a. ”Upper and lower bounds on the rate of convergence for nonhomogeneous birth and death processes.” Stoch. Proc. Appl. 59, 157–173.

ANNA KOROTYSHEVA is Candidate of Science (PhD) in physics and mathematics, associate professor, Vologda State University. Her email is [email protected]. KSENIA KISELEVA is PhD student, Department of Applied Mathematics, Vologda State University. Her email is [email protected]. VICTOR KOROLEV is Doctor of Science in physics and mathematics, professor, Head of Department of Mathematical Statistics, Faculty of Computational Mathematics and Cybernetics, M.V. Lomonosov Moscow State University; leading scientist, Institute of Informatics Problems, Federal Research Center ”Computer Science and Control” of the Russian Academy of Sciences. His email is [email protected].