Information Theoretic Decomposition of GE-Type Closed Queueing

Information Theoretic Decomposition of GE-Type Closed Queueing Networks with Finite Capacity and Multiple Servers

Vassilios D. Tsiantos* and Demetres D. Kouvatsos†

Abstract. In this paper, the information theoretic principle of Minimum Relative Entropy (MRE), given fully decomposable subset and aggregate mean value constraints, is applied, in conduction with classical queueing theory, to extend earlier works and derive new analytic approximations for the conditional and marginal state probabilities of arbitrary closed queueing network models (QNMs-B) with Generalised Exponential (GE) multiple server stations and RS blocking. A generalised MRE decomposition algorithm is developed in the context of a multilevel variable aggregation scheme and is based on asymptotic connections to infinite capacity multiple server queues and related closed-form expressions for the determination of the effective and overall flows in the network. The GE/GE/c/k;N censored queue with GE-type interarrival-time and service-time distributions, multiple servers, minimum queue length k and finite capacity N, is used as the building block in the solution process. Numerical results are included to illustrate the credibility of the proposed MRE decomposition algorithm against simulation.

1. Introduction Networks of queues with finite capacities are widely recognised as useful and powerful tools in representing more realistic models of discrete flow systems such as computer systems, telecommunication networks and flexible manufacturing systems. Blocking arises because the flow of customers through one queue may be momentarily stopped if the destination queue has reached its capacity. Various types of blocking mechanisms have been considered in the literature. One of the most important mechanisms is that of repetitive service (RS) blocking with either fixed (RS-FD) or random (RS-RD) destination. This kind of blocking occurs when a customer upon service completion at queue i attempts to join a destination queue j whose capacity is full. Consequently, the customer is rejected by queue j and immediately receives another service at queue i. In the case of RS-FD blocking this is repeated until the customer completes service at queue i at a moment where the destination queue j is not full. In the RS-RD case each time the customer

*

Electronic & Information Storage Systems Research Group, University of Manchester, School of Engineering, Manchester, Oxford Road, M13 9PL, UK. Email: [email protected] † Computer Systems Modelling Research Group, University of Bradford, Computing Department, Bradford BD7 1DP, West Yorkshire, UK. Email: [email protected]

completes service at queue i, a downstream queue is selected independently of the previously chosen destination queue j. Kouvatsos and Xenios [Kou89a, Kou89b], and Kouvatsos and Denazis [Kou93] obtained in the context of maximum entropy (ME) formalism, product-form approximations for general open and closed queueing networks with arbitrary configurations by assuming RS blocking and GE (generalised exponential) type interarrival-time and service-time distributions. The main aim of this paper is to propose a new analytic methodology for the analysis of general closed QNMs-B with multiple servers, based on the concept of hierarchical system decomposition and the principle of MRE. The principles of maximum entropy (ME) and minimum relative entropy (MRE), a generalisation, are uniquely correct, self-consistent methods of inference for estimating probability distributions based on information in the form of expected values [Sho80, Sho81]. System decomposition is one of the most interesting and challenging techniques in analysing QNMs [Cha75, Cou77]. Decomposition methods proceed in two major steps: i. analysis of subsystems, and ii. analysis of a macro system composed of these subsystems using the results of step i. Courtois [Cou77] first investigated this approach in the analysis of closed queuing networks. This decomposition scheme proceeds formally in a hierarchical multilevel fashion and as result there is a precise set of conditions for determining how and when decomposition is viable. At each level of decomposition, a closed two-station cyclic queueing system is analysed. This cyclic system is composed of the subsystem of the previous level acting as a composite server with load dependent service rates and the next most strongly connected station. The ME principle have been used by Kouvatsos and Tomaras [Tom91, Kou92] in order to derive new analytic approximations for the conditional and marginal state probabilities of single class general closed QNMs without blocking in the context of a multilevel variable aggregation scheme. In this paper the MRE principle is used analytically, via asymptotic connections to infinite capacity queues, in order to decompose hierarchically the equilibrium state probabilities of a general closed QNMs-B with single class customers. The point of departure is taken to be the work of Shore [Sho82] on an abstract system using MRE given fully one-level decomposable constraints which are either expectations conditional on a specific subset or expectations involving aggregate subset (marginal) probabilities. Shore [Sho82] has shown via special properties that the overall MRE problem is solved by determining each of the MRE conditional subset probability distributions, followed by the MRE solution for the aggregate distribution. In an earlier work the authors used the M/M/1/k;N queue with exponential interarrival-time and service-time distributions, single server, minimum queue length k and finite capacity N as the building block in the decomposition process [Tsi97]. Moreover, the GE/GE/1/k;N queue with GE-type interarrival-time and service-time distributions, single server, minimum length k and finite capacity N has been used by the authors as the building block in the solution process in [Tsi98]. In this paper the GE/GE/c/k;N Censored Queue, with GE-type interarrival-time and service-time distributions, multiple servers, minimum queue length k and finite capacity N, is used as the building block. A review of the ME solution of GE/GE/c/k;N Censored Queue [Kou89b] is presented in Section 2. In Section 3, the MRE methodology for queueing networks with blocking is described analytically, while in Section 4 the credibility of the algorithm is examined. Conclusions and further work follows in Section 5.

2. The GE / GE / c / k ; N Censored Queue Consider a FCFS multiple-server queue having GE-type interarrival-time and service-time distributions with known first moments and finite capacity N. It is assumed that the arrival and departure processes are “censored”, i.e. arriving customers are turned away when buffer N is full and departures are not allowed to occur from the state k, 0≤ k i=l∑+1 Ni  L - i=l  , i=1,...,M-1, and βi =  M   0, L ≤ ∑ N i i=l+1 

βM =

M-1

∑N

, i=M, and γ i = min(L, i

i=0

l

∑ N ). i

i=0

3.1 A MRE decomposition algorithm for a general closed QNM-B For exposition purposes consider a central server model with two I/O units. 3.1.1 The state space The state space of this system is defined by D = { n=(n0, n1, n2) / ki ≤ ni ≤ Ni, i = 0, 1, 2, and n0 + n1 + n2 = L}, the set of joint queue length states of the three queues involved. The partition of the state space D is { Dk , D k +1 , ..., D N } with 2

2

2

Dn2 = { n=(n0, n1, n2) / k0 ≤ n0 ≤ min(L-n2,N0), k1 ≤ n1 ≤ min(L-n2,N1), and n0 + n1 = L - n2 }. The partition of the state space is a single-level one. Each subset Di contains all possible states of the system assuming that i customers are present at the second I/O (unit Σ2) or L-i customers circulate in the subsystem of the first level, which consists of the CPU and the first I/O (unit Σ1). The joint queue length probability distribution is given by {P(n0, n1, n2), (n0, n1, n2)} and the conditional distributions by { P1(n0 / n2), (n0 = k0, k0+1, ..., N0-n2) } , n2 = k2, k2+1, ..., N2 , (3.1)

P( n0 , L - n 0 − n2 , n 2 ) Pr{ D n 2 }

P1(n0 / n2) =

where

(3.2)

and Pr{ D n } is the probability that the state of the system is one of the elements of 2 subset D n . This aggregate distribution is actually the marginal queue length 2 distribution of the second I/O. Hence, Pr{ D n } =P2(n2), n2=k2,k2+1,...,N2. 2

(3.3)

Thus, if we specify distributions (3.1) and (3.3) we can easily derive the joint probability distribution using relation (3.2). Moreover, we may derive the marginal queue length distributions for queues Σ0 and Σ1 using the law of total min( L − n 0 , N 2 )

probability as

P0 ( n 0 ) =

∑ P (n 1

0

/ n 2 ) P2 ( n 2 ) , n0=k0,k0+1,...,N0,

(3.4)

n2 =k 2 min( L − n 1 , N 2 )

and P1 (n 1 ) =

∑ P (L − n 1

1

− k1 / n 2 ) P2 ( n 2 ) , n1= k1, k1+1, ..., N1.

(3.5)

n2 =k 2

3.1.2 Constraint Information 3.1.2.1 Subset Constraints In the first level of aggregation the conditional distribution (3.1) depends only upon interactions between Σ0 and Σ1. For each subset D n , n2 = k2, k2+1, ..., 2

N2-1, we have the following constraints •

L−n2

∑ P (n

the normalisation

1

0

/ n2 ) = 1

(3.6)

n0 =k0 k1 ≤ n1 ≤ N 1

•

the conditional mql excluding j jobs, < n 0 - j > L − n 2 ( < L-n2 ) min(L − n 2 , N 0 )

∑

f 1,1 (n 0 ) P1 (n 0 /n 2 ) = < n 0 - j > L − n 2

n0 = k 0 k1 ≤ n1 ≤ N 1

(3.7)

where f1,1(n) = max(0,n-j), ∈ (0,N), and j = max(c0,k0+1). • the conditional probabilities < U 0 (l 0 ) > L − n written as: 2 min(L − n 2 , N 0 )

∑

f 1,2 (n 0 ) P1 (n 0 /n 2 ) = < U 0 (l 0 ) > L − n 2

(3.8)

n0 = k 0 k1 ≤ n1 ≤ N 1

0, if n 0 < l 0 . 1, otherwise

where f 1,2 =  •

the conditional fb equation, P1(L-n2/n2) = < Φ0 > L − n , 0< < Φ0 > L − n < 1 as 2 2 min(L − n 2 , N 0 )

∑

f 1,1 (n 0 ) P1 (n 0 /n 2 ) = < Φ0 > L − n ,

(3.9)

2

n0 = k 0 k1 ≤ n1 ≤ N 1

where f1,3(n0)=max{0,n0-min(L-n2,N0)}. 3.1.2.2 Aggregate constraints In the second and final level of aggregation it is assumed that for the marginal state distribution { P2(n2), n2 = k2, k2+1, ..., N2-1, N2 } the following aggregate constraints exist: •

min( L , N 2 )

the

∑ P (n

normalisation

2

2

) = 1

n 2 =k2

(3.10) • the marginal mql excluding j jobs, < n 2 - j > L ( L

(3.11)

n2 =k2

•

the marginal probabilities < U 2 (l 2 ) > L ( < < n 2 > L ) written as min(L,N 2 )

∑

f 2,2 (n 2 ) P2 (n 2 ) = < U 2 (l 2 ) > L

(3.12)

n2 =k2

0, if n 2 < l 2 . 1, otherwise

where f 2,2 =  •

min(L,N 2 )

the marginal fb equation,

∑

f 2,3 (n 2 ) P21 (n 2 ) = L

(3.14)

n2 =k2

where f2,3(n2)=max{0,n2-min(L,N2)+1}.

3.2.1 The MRE solution 3.2.1.1 First level of aggregation In the first level of aggregation, the MRE solution, subject to constraints (3.6)-(3.10) is

P1 ( n 0 / n 2 ) = p(k 0 ) x 01,1 f

(n 0 )

f

g 01,2

( n0 )

f

y 01,3

( n0 )

,

(3.15)

where n0=k0,k0+1,...,min(L-n2,N0), p(k0) is given by formulas (2.2) or (2.3), ρ0 is the asymptotic utilisation of unit Σ0 in this first level and is defined as

!0 =

c1  1 , c0 0

while x0, g0, y0 are given by the formulas (2.9), (2.7)-(2.8) and (2.10), respectively. The prior to be used at the second level of aggregation is denoted as {G(1,L-n2), n2=k2,k2+1,...,N2 }, and given by min(L − n 2 , c1 )

∏

G(1,L-n2)=

g 1, j

j =1

P1 (k 0 /L − n 2 )

, (3.17a) if ρ0≤1 and

min(L − n 2 ,c1 )

∏

G(1,L-n2)=

j=1

g 0, j

P1 (L − n 2 /L − n 2 )

, if ρ0>1.

(3.17)

3.2.1.2 Second level of aggregation In the second and final level of aggregation, applying the marginal constraints (3.11) - (3.15) on the prior defined by (3.23), the MRE solution is,

P2 (n 2 ) = G(1, L - n 2 ) x 22,1 f

(n 2 )

f

g 22,2

(n 2 )

f

y 22,3

(n 2 )

, (3.18) where n2=k2,k2+1,...,N2, and x2, g2,j are given by formulas (2.9) and (2.7)-(2.8), respectively, and y2 = L-n  min( L − n 2 , c2 ) − 1 2  kµ P ( n ) ∑ U ( k , j)P ( n ) ∑ i 2 2 i,1 2 2  k=k j= k 2  1 ∑  + µ i min(L - n 2 , c i ) i =0  L- n 2  ∑ U i,1 (min(L - n 2 , c i ), L − n ) P2 ( n 2 − 2  j= min(L - n 2 ,ci )  c µ P2 ( n 2 ) 2

  c 2 − 1    ∑ kµ P ( k )  2 2  = k 1    ri2( 2 ) -   L-n −1  2 + c µ  ∑ P ( k) j)  2 2 k=c 2     2 

2

4. MRE Algorithm for general closed QNMs-B The ME solution in case of GE-type networks is

P(n ) =

1 M n i − k i f(n i ) h(n i ) ∏ xi gi yi , Z i=0

(4.2)

where xi, gi, and yi are given by formulas (2.2), (2.3) and (2.4). Blocking probabilities πci and effective routing probabilities rij are given by πci = rij π ij and rij =

rij (1 - π ij ) (1 − π ci )

∑

, respectively.

4.1 Procedure FLOWITERATION(l) Procedure FLOWITERATION(l) is called at the (l+1)th level of decomposition. It is applied iteratively for the asymptotic flow parameters of the subsystem (Σ0 Σ1 ... Σl) in order to be calculated the asymptotic utilisation of the station Σl+1. Step 1 Prepare the asymptotic utilization to be used. Step 2 Initialize λ ij Step 3 Calculate blocking probabilities πci . Step 4 Calculate effective routing probabilities, rij(l) . Step 5 Solve effective job flow balance equations Step 6 Calculate effective service parameters, { µi }, i=0, 1, 2, ..., M. Step 7 Calculate blocking probabilities πci . Step 8 Calculate overall actual parameters, {Λi,}, i=0, 1, 2, ..., M, Step 9 Prepare network parameters according to feedback transformation on R(l+1).  and input rate λl+1 of unit Σl +1. Step 10 Evaluate asymptotic utilization ρ l +1 Note that in step 5 of the above procedure formula (4.1) was used, since at (l +1) , because the feedback transformation that stage µl+1 has the value µ l*+1 ( 1 - rl+1,l +1 ) has been applied. We also have to note that the routing matrix R(l) was used in the iteration, while the links between the subsystem (Σ0 Σ1 ... Σl) and unit Σl+1 were specified by the last column of matrix R(l+1) as it should be done, according to the asymptotic form (4.1) of the flow balance equation,

U l (l , N l ) µ l [1 - rll(l) ] =

l-1

∑µ

k

rkl(l) U k ( l , N l ) .

(4.4)

k =0

4.2 MRE decomposition algorithm for GE-type closed QNMs-B Input { M, L, Ni , rij(M ) , i,j = 1, 2, ..., M } STEP Step Step Step Step Step Step Step STEP Step Step Step Step STEP Step Step Step

1 ( 1st level of aggregation) 1.1 Calculate routing matrices R(1), R(2), ..., R(M-1). 1.2 Apply feedback transformation. 1.3 Calculate ρ0 , ρ1 . 1.4 Calculate x0 ( GE/GE/c/k0 ; N0 Censored Queue ). 1.5 Evaluate conditional distributions P1(k1/N1), P1(n1/N1), P1(N1/N1). Step 1.5.1 Calculate G*(1, N1 ). 1.6 Evaluate relative bottleneck and prior. 1.7 Evaluate conditional utilisations of Σ0 , Σ1 . 2 (lth level of aggregation) 2.1 FLOWITERATION(l) 2.2 Calculate xl.. 2.3 Evaluate conditional distributions Pl (kl /Nl ),Pl (nl /Nl ),Pl (Nl /Nl ). Step 2.4.1 Calculate G*(l, Nl.). 2.4 Evaluate relative bottleneck and prior. 3 (Calculate the marginal distributions) 3.1 Calculate marginal distributions PM ( n ), PM* ( n ). 3.2 Calculate marginal distributions Pl ( n l ), Pl* ( n l ). 3.3 Calculate P0 (n 0 ), P1 (n 1 ).

4.3 Discussion When a decomposition method, based on a hierarchical multilevel partition of the state space, is used to analyse approximately a QNM, the resulting solution is not unique, but depends on the order that the stations of the network are joined together and thus effectively on the enumerations of the stations. The decision on the enumeration of the stations is based on the interaction rates at every level of aggregation. In fact, in the first level of aggregation, an average interaction rate is associated to every pair of stations (Σi,Σj), i,j = 0, 1, ..., M, which is given by, µ *i rij( M ) + µ *j rji( M ) . Thus the pair of stations to be joined at the first level is the one with maximum interaction rate. At level l of decomposition (l>1), given that the subsystem (Σ0, Σ1, ..., Σl-1) has been defined, an average interaction rate between stations Σj, j = l, l+1, ..., M and the subsystem is associated and given by µ *j

l -1

∑ i=0

l -1

rji( M ) +

∑µ i=0

* (M) i ij

r

. So the station to be joined to

the subsystem (Σ0, Σ1, ..., Σl-1) is the one with maximum interaction rate. 4.4 Numerical Results In this section the credibility of the MRE algorithm is illustrated against simulation (SIM) involving typical QNMs-B with central server and arbitrary configuration networks. SIM results are generated by using the queueing network analysis package QNAP-2 [Ver84] at 95% confidence intervals. Note that the deviation of these intervals from the central value of simulation is, generally, less than 1%. Numerical results are randomly chosen from an exhaustive experimentation carried out by the authors and are displayed in Table 1 and Figures 1-2. Relative comparisons against simulation are based on the mean queue lengths {}, and marginal state probabilities {Pi(ni), ni = ki, ..., Ni}, i=0, 1, ..., M. For validation purposes the following error measures (EM) are used: i) EM for the mean queue lengths defined by the absolute ratio:

EM(< n i >) =

SIM( < n i >) - MRE( < n i >) , i=0,1,2, ..., M M

∑ SIM(< n

j

>)

j= 0

ii) EM for the marginal state probabilities, {Pi(ni)}, defined by the absolute difference: EM(Pi(ni)) = | SIM(Pi(ni)) - MRE(Pi(ni) |, ni= ki, ..., Ni , i=0,1,2, ..., M. The MRE algorithm is said to be within a tolerance ε (ε>0), if the estimated EMs for the aforementioned marginal statistics do not exceed ε for all i=0,1,2, ..., M. Note that in our investigation, the accuracy of MRE algorithm is considered to be good, if tolerance is less than 0.05 and adequate if tolerance ε ∈ [0.05, 0.1). More details on the definition of EMs and suitability of tolerance levels can be seen in Chandy et al [Cha75]. From Table 1 and Figures 1-2 can be observed that the MRE results are very comparable in accuracy to those obtained via simulation models.

4.4.1 Computational cost In general, algorithms that are based on hierarchical decomposition schemes are computationally more expensive than algorithms based on convolution techniques, but overall they are comparable [Cou77]. The problem that is of major importance in decomposition algorithms is their comparatively large memory requirements. Note that in order to obtain the marginal queue length distributions, an M×(Ni-ki)2 size array has to be used for the conditional distributions, as well as an M2×(Ni-ki) size one for the conditional utilisations. In case of networks with small number of stations (M) but large population of customers (L), some space may be saved at the cost of determining only the marginal utilisations and mean queue lengths, without calculating the marginal distributions. MRE algorithm needs much less time to run than simulation, and is very comparable to the ME convolution algorithm proposed by Kouvatsos and Denazis [Kou93] which is of O(mML), where m is the number of iterations for applying the convolution type procedure in order the job flow balance equations of the closed QNM to be satisfied.

5. Conclusions and further work Relative entropy minimisation given fully decomposable subset and aggregate constraints in conjunction with asymptotic connections to infinite capacity queues, provides a new framework for the analytic approximation of the conditional and state probabilities of general closed queueing networks with blocking under repetitive service (RS) mechanism in the context of a multilevel variable aggregation scheme. The concept of subparallelism is used to preserve the flow conservation. Numerical results based on GE-type networks demonstrate the credibility of the MRE decomposition algorithm. Extension of the work to the case of exponential closed QNMs-B with multiple servers is the subject of current study.

References [Cha75a] K.M. Chandy, U. Herzog, and L. Woo. Parametric Analysis of Queueing Networks, IBM J. Res. Develop, January 1975, 36-42. [Cha75b] K.M. Chandy, U. Herzog, and L. Woo. Approximate Analysis of General Queueing Networks, IBM J.Res.Devel., Jan. 1975, 43-49. [Cou77] P.J. Courtois. Decomposability - Queueing and Computer System Applications, Academic Press, New York, 1977. [Kou85] D.D. Kouvatsos. Maximum entropy methods for general queueing networks, In Mod. Tech and Tools for Perf. Anal. (D. Potier, ed.), 589-608, NorthHolland, 1985. [Kou89a] D.D. Kouvatsos, and N.P. Xenios. Maximum Entropy Analysis of General Queueing Networks with Blocking, in Queueing Networks with Blocking (H.G. Perros and T. Altiok, eds.), 281-309, North Holland, Amsterdam, 1989. [Kou89b] D.D. Kouvatsos, and N.P. Xenios. MEM for Arbitrary Queueing Networks with Multiple General Servers and Repetitive-Service Blocking, Performance Evaluation 10, 1989, 169-195.

[Kou92] D.D. Kouvatsos and P.J. Tomaras. Multilevel aggregation of central server models: a Minimum Relative Entropy approach, Int. J. Systems Sci., 23(5), 1992, 713-739. [Kou93] D.D. Kouvatsos and S.G. Denazis. Entropy maximised Queueing Networks with Blocking and Multiple Job Classes, Perf. Eval. 17, 1993, 189-205. [Sho80] J. E. Shore and R. W. Johnson. Axiomatic Derivation of the Principle of Maximum Entropy and the Principle of Minimum Cross-Entropy, IEEE Trans. Inform. Theory, IT-26(1), 1980, 26-37. [Sho81] J. E. Shore and R. W. Johnson. Properties of Cross-Entropy Minimization, IEEE Trans. Inf. Theory, IT-27(4), 1981, 472-482. [Sho82] J. E. Shore. Cross-Entropy Minimization given Fully Decomposable Subset and Aggregate Constraints, IEEE Trans. Inf. Theory, IT-28(6), 1982, 956961. [Tom91] P.J. Tomaras and D.D. Kouvatsos. MRE hierarchical decomposition of general queueing network models, Acta Inform.,28, 1991, 265-295. [Tsi97] V.D.Tsiantos and D.D.Kouvatsos. Minimum Relative Entropy (MRE) Hierarchical Decomposition of Closed Queueing Networks with Repetitive Service (RS) Blocking, Proceedings of the 13th UK Performance Engineering Workshop, 26/1-26/16, 1997, Ilkley, UK. [Tsi98] V.D.Tsiantos and D.D.Kouvatsos. Information Theoretic Decomposition of GE-Type Closed Queueing Networks with Finite Capacity, Proceedings of the 14th UK Performance Engineering Workshop, 1998, 133-149, Edinburgh, UK. [Van78] H. Vantilborgh. Exact Aggregation in Exponential Queueing Networks, Journal of the ACM, 25(4), 1978, 620-629. [Ver84] M. Veran, and D. Potier. A Portable Environment for Queueing Systems Modelling, Rep. 314, INRIA, 1984.

Table 1 Central server model. Input data: M=3, L=11, a01=0.4, a02=0.6, a10=1, a20=1, the rest routing probabilities are 0, µi=2,6,3, C2=2, Ni=5, ci=3.

Performance statistics

P[1] P[2] P[3] P[4] P[5]

Queue SIM 0.0011 0.0127 0.0289 0.1208 0.8364 4.7790

1 MRE 0.0021 0.0162 0.0393 0.2359 0.7065 4.6285

Queue SIM 0.5134 0.3074 0.0996 0.0577 0.0219 1.7670

2 MRE 0.4822 0.3454 0.0887 0.0593 0.0244 1.7983

Queue SIM 0.0099 0.0399 0.0717 0.2436 0.6349 4.4540

3 MRE 0.0116 0.0345 0.0665 0.2247 0.6627 4.4924

0.14 0.12 0.1

P[1]

0.08

P[2]

0.06

P[3]

0.04

P[4] P[5]

0.02 0 1st Queue

2nd Queue

3rd Queue

Figure 1. Bar-chart of EM of marginal probabilities for central server model (cf. Table 1).

0.014 0.012 0.01 0.008 0.006

mql

0.004 0.002 0 1st Queue

2nd Queue

3rd Queue

Figure 2. Bar-chart of EM of mean queue lengths (cf. Table 1).