Performance analysis of interprocessor communications in an electronic switching system with distributed control

83

Performance Analysis of Interprocessor Communications in an Electronic Switching System with Distributed Control* Shuichi Sumita NTT Communication Switching Laboratories, 9-10, Midori-Cho 3-Chome, Musashino-Shi, Tokyo 180, Japan

Received March 1988 Revised August 1988 and October 1988

Interprocessor communications as well as interconnection among processors is an important design factor in an electronic switching system with distributed control. This paper proposes a new scheme for a message transfer algorithm in interprocessor communications, called a gating scheme. A major difference from the usual clocked scheme is that a processor never requests message transfer to another processor until it has finished processing all tasks in that processor; in the clocked scheme a processor periodically sends requests to another processor. A queueing model is presented for the performance evaluation of the gating scheme. Various performance measures are derived using the theory of piecewise Markov process. The gating and clocked schemes are compared numerically. Kevwords: Distributed Control Switching System, Interprocessor Communications, Queueing System with a Switch, Gating Scheme, Piecewise Markov Process.

Shuichi Sumita received his B.E., M.E., and Ph.D. degrees in Applied Mathematics from Kyoto University in 1975, 1977, and 1988, respectively. He joined NTT Laboratories in 1977. Since then, he has worked on performance evaluation of a variety of communications systems developed by NTT, including digital switching systems. He is currently Supervisor in the Communication Switching Laboratories. His research interests include distributed switching system architecture, ATM networks, performance evaluation of communications systems, queueing theory, and simulation.

* This paper was presented at the 2nd International Workshop on Applied Mathematics and Performance/Reliability Models of Computer/Communication Systems, which was held at Rome in May 25-29, 1987. North-Holland Performance Evaluation 9 (1988/89) 83-91

1. Introduction

R e c e n t l y developed electronic switching systems have a d o p t e d d i s t r i b u t e d control architecture. This architecture is used because of various advantages such as m o d u l a r software i m p l e m e n t a tion, high-reliability a n d linear growth facilities. These a d v a n t a g e s c a n n o t be easily realized through centralized c o n t r o l architecture. I n most of the systems with d i s t r i b u t e d control architecture, call processing functions, such as detection of call origination, receiving digits, translation of digits a n d speech path selection, are hierarchically d i s t r i b u t e d over two or more groups of processors. A typical example is a system that consists of two groups of processors; one group of processors performs b o t h event detection a n d sign a l i n g f u n c t i o n s a n d the other performs call control function. Here, the former is called a signaling processor a n d the latter is called a call control processor. I n such a f u n c t i o n sharing system, the i n t e r c o n n e c t i o n b e t w e e n signaling a n d call control processors is designed to fully utilize the p o t e n t i a l power of d i s t r i b u t e d c o n t r o l architecture. Thus, there have b e e n n u m e r o u s studies c o n c e r n i n g what type of topology should be chosen for interconn e c t i o n ; interesting a n d useful results have been o b t a i n e d by m a n y authors (for example, see [19]). Interprocessor c o m m u n i c a t i o n s between two groups of processors is also i m p o r t a n t because it specifies a message transfer scheme b e t w e e n two groups of processors for a given i n t e r c o n n e c t i o n topology. A message transfer scheme is directly related with a delay i n c o m m u n i c a t i o n s from a signaling processor to a call control processor or from a call c o n t r o l processor to a signaling processor. Since interprocessor c o m m u n i c a t i o n s delay is a m a j o r c o m p o n e n t of call setup delay, a long delay in interprocessor c o m m u n i c a t i o n s leads to a long call setup delay. This p a p e r therefore focuses on a p e r f o r m a n c e analysis of interprocessor c o m m u n i c a t i o n s .

0166-5316/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)

84

S. Sumita / Performance analysis of interprocessor communications Signaling processor

IPC

Call

control

processor

Q1

IPC:Interprocessor Communications Controller Fig. 1. Queueingmodel for interprocessorcommunications.

Interprocessor commumcations can be realized as message transfer between signaling and call control processors. A simple queueing model representing this is shown in Fig. 1 [16]. A message originating in a signaling processor enters queue Q1 and waits for the switch to close. When the switch closes, messages in Q1 move to Q2 in a call control processor. Here, note that the algorithm in closing and opening the switch is related with interprocessor communications. It is essential to the design of the interprocessor communications system to determine what algorithm to use. A type of queueing system with a switch was first proposed to evaluate the messsage transfer algorithm between a peripheral device and a central control unit in an electronic switching system with centralized control [14,15]. The class of queueing model shown in Fig. 1 has recently aroused interest because it can describe message transfer schemes between processors in a distributed control system [16-18,25,26]. A common algorithm is the clocked scheme in which the switch closes and opens at a fixed interval. If this scheme is adopted as message transfer scheme in the interprocessor communications, the length of time between two successive instants at which the switch closes (scan interval) should be determined to minimize the effect of the overhead associated with the closing operation of the switch. Thus, various queueing models have been proposed and interesting results concerning how to optimize the scan interval have been found [25]. References [17,18,26] analyzed more complicated schemes than the clocked scheme. However, if the overhead associated with closing and opening the switch significantly affects system performance, another algorithm with less overhead may be necessary. This paper proposes another algorithm for closing the switch, called a gating scheme, to reduce the effect made by the overhead associated with

the closing operation of the switch [21]. In the gating scheme, a processor never closes the switch until it has finished processing all messages in Q1. This is a major difference compared-to the clocked scheme. The queueing system considered in this paper is similar to cyclic service systems [1-11,20,22-24, 27]. In fact, the queueing model shown in Fig. 1 is a cyclic service system with node dependent overheads as analyzed by Ferguson [9]. Ferguson analyzed an exhaustive scheme for a multiple queue system and derived the generating function of the waiting time distribution and approximations for the mean waiting time. This paper analyzes a gating scheme for a single queue system, i.e. a simpler case, and derives an exact result for the mean waiting time. This paper is organized as follows. Section 2 describes the gating scheme and points out the similarity of the system under consideration to cyclic service systems. Section 3 derives the mean dealy for the gating scheme using the theory of piecewise Markov process. Section 4 compares the gating scheme with the clocked scheme in terms of the mean delay.

2. Queueing model This section describes the queueing model shown in Fig. 1 in detail. In this system, customers arrive at Q1 according to a Poisson process with rate X and join it to wait for service. Switch SW is located between queues Q1 and Q2 to control the flow of customers from Q1 to Q2. The number of waiting places in each queue is infinite. When the switch closes, all the customers in Q1 can move to Q2. A scheme for closing and opening the switch will be explained later. After customers have moved to Q2, they wait for service in Q2. Customers in Q2 are served on a first-come first-

S. Sumita / Performance analysis of interprocessor communications

L

Time'L-

is

dependent E v E v

_L

on

the

number

of

customers

= nh i f Y ( t n - ) ~ n , where = T - d if Y(tn-) = O.

85

d _1

in

Q1 j u s t

before

tn,Y(t~-).

n > O.

Fig. 2. Scan interval.

served basis. Their service times are i.i.d, random variables with distribution function H(x). Its mean and second moment, denoted by h and h (2), are finite, respectively. The LST of H(x) is denoted b y / ~ ( s ) . To describe the model completely, it is necessary to explain an algorithm for closing and opening the switch. In addition to the clocked scheme, another algorithm, called the gating scheme, is proposed. The gating scheme can be described in the following way. In this scheme, the server never closes the switch until it has finished serving all the customers in Q2. When the server has finished serving customers in Q2, additional time (overhead) is taken to close the switch. If there are customers in Q1 when the switch closes, all the customers in Q1 will instantly move to Q2 and the switch will open. It is assumed that the transfer time of a customer from Q1 to Q2 is short compared to service time and overhead time. Thus, the transfer time will be neglected in the analysis. If there are no customers in Q1, the switch immediately opens, and it will close again after a fixed interval of time T. Thus, the switch closes every interval of time T until at least one customer is found in Q1. We can see that the clocked scheme is used while the server finds no customers waiting in Q1. The overhead time required to close the switch is constant time d. The interval of time between two successive instants at which the switch closes, referred to as scan interval, is dependent on the number of customers in Q1 just before the switch closes. If the number of customers in Q1 just before the switch closes is n > 0, then the mean scan interval is (nh + d). If there are no customers in Q1 just before the switch closes, then the mean scan interval is T. Fig. 2 shows the scan interval.

For convenience, the distribution function of the overhead time is abbreviated as D(x). That of a scan interval when there are no customers waiting in Q1 is abbreviated as T(x). The LSTs of them are denoted by D ( s ) and 7~(s), respectively. D ( s ) and 7~(s) are given by /)(s) = exp(-sd) and T(s) = exp(-sT). The traffic intensity is denoted by p (=)~h). The queueing system considered here is a special case of a cyclic service system with node dependent overheads. If T = d, however, the overhead is node independent and thus the model is the same as the cyclic service system with gated service [11]. Finally, the relevance of the queueing model considered here to real systems, especially switching control systems, is described below. Switch SW symbolizes the message transfer scheme of the IPC. That is, a transfer algorithm can be represented by an algorithm for closing and opening the switch. In a switching control system, scanning trunks and collection of digits are tasks having strict timing requirements. The clocked scheme is used to control the execution of such a task. A major reason for the use of that scheme is to shorten the delay for acting upon the external stimuli such as digits. In an interprocessor communications, the total delay as well as the delay in Q1 or Q2 is important, where the total delay is the sum of the delays in both Q1 and Q2. To determine what algorithm to use, it is necessary to compare the gating scheme with other schemes in terms of the mean total delay.

S, Sumita / Performance analysis of interprocessor communications

86

3. Analysis

3.2. Solution of the functional equation

This section derives the mean waiting time in each queue and the mean total delay using the theory of the piecewise Markov process [13].

The functional equation derived in Subsection 3.1 is the equation treated by Kuczma [12]. A solution of this functional equation can be derived by an iteration procedure. First, let us introduce a sequence of functions fn(z) defined as

3.1. Problem formulation Let Y(t) denote the number of customers in Q1 at time t. Also, let tn be the n th instant when the switch closes (n >t 1) and y. be the number of customers in Q1 just before 6 , that is, y, = Y(t,-). Then, the stochastic process (yn } is an imbedded Markov chain of Y(t). The steady state probability of this Markov chain { u i } is defined as

fo(z)=z

for Iz[ ~<1,

n=0,1,2,

f~+l(z)=H(X(1-f~(z))),

....

Substituting f,(z) into z in both sides of (4), and using the above relationships, we have = v(L+,(z))b(x(1

-L(z)))

-fo(z)))

u i= lira P r ( y , = i ) . ~ ---* oo

Assuming that a steady state exists, we have the following linear system of equations:

- b(h(1-f.(z)))].

(5)

Applying (5) iteratively, we have n--1

Ui = ~

i >~O,

ujpji ,

(1)

U(z) = U ( f . ( z ) ) I-I /T}(X(1 - f k ( z ) ) )

j=O

k=O

where pj~ (i >/O, j >1 O) is the one-step transition probability of the Markov chain, which is defined

+ u0(7~(X(1 - z)) -/T) (X (1 - z ) ) )

as

+u o y' (7~(X(1-fk(z)))

n--1

Pj,= P r ( y , + l = i l y , = j ) .

k=l

-

For j >~ 1, Pji is given by k-1 =

× ]--I D ( X ( 1 - - f m ( Z ) ) ) .

(6)

m=0

exp(-kx) d{D*H*J(x)},

i>~O.

(2)

For j = 0, Poi is given by oo

i

Poi=fo { ( X x ) / i ! ) e x p ( - h x ) d T ( x ) ,

i>~O, (3)

where the convolution of the distribution functions D(x) and H(x) is denoted by D * H(x) and j-fold convolution of H(x) is denoted by H * J ( x ) . To solve the linear system of equations (1), we can use the generating function of the stationary distribution ( u i } defined as u(z) =

u,z'. i=O

From (1), (2) and (3), we have the following functional equation of U(x):

U(z) = U(/4(X (1 - z)))/T}(X (1 - z))

+Uo(f'(X(1-z))-D(X(1-z)) ).

(4)

To ensure that all terms depending on n in (6) converge to a finite value as n increases to infinity, the following three lemmas will be necessary: 3.1. Lemma. The sequence {f~(x)} uniformly con-

verges to 1 if p < 1. Proof. Let us introduce the following function g ( x ) defined on [0,1]: 8(x) = x -

-x)).

Differentiating g(x) twice, we have 8 ' ( x ) = 1 + XB'(X(1 - x)) and 8 " ( x ) = -X2/-I"(~(1 - x)). For 0 < x < 1, g"(x) < 0. Thus, g'(x) is a monotone decreasing function on [0,1]. Furthermore, g'(0)=l+h/-I'(X)>0 and g ' ( 1 ) = l - p > 0 . Thus, 8'(x) > 0 for x in [0,1] and hence g(x) is a monotone increasing f u n c t i o n . - S i n c e g(0) = - / 4 ( h ) < 0 and g(1) = 0, we have g(x) < 0 for 0 < x < 1. Substituting fn(x) into g(x), we have

87

S. Sumita / Performanceanalysis of interprocessor communications

f~(x)-/4(X(1-f~(x)))<0, Thus, it has been shown that f , ( x ) O"-lx + 1 - O~. According to the Taylor expansion, / ~ ( X ( 1 - x))

= fl(0) - X(x - 1)/1'(0) + X2(x - 1)2/-I"(X(1 b))/2, where b is some value that satisfies x < b < 1. Thus, fl(),(1 - f ~ ( x ) ) ) > 1 + O ( £ ( x ) - 1). Thus, we have

3.3. Lemma. F I T , - I D ( X ( 1 - f k ( x ) ) ) converges if P <1. Proof. Since D ( X ( 1 - f k ( x ) ) ) < 1 and f k ( x ) converges if p < 1, it is clear that [I~ Ib(X(1 - f k ( x ) ) ) converges. [] From Lemmas 3.1-3.3, we see that the steady state exists if 0 < 1. In such a case we can take the limit with respect to n in (6). The following equation is thus obtained: ~o

f ~ + i ( x ) > p £ ( x ) + l - o.

u(.)

Using this inequality iteratively, we have f , ( x ) >O"x + 1 - f

= FI b(x(1 n=0

+ Uo(7~(X(1 - z)) - D(),(1 - z ) ) )

> 1 -p".

oo

Letting n --, oc in the above inequality, 1 ~< lim . . . . f , ( x ) . Since f , ( x ) ~< 1, it follows that the sequence uniformly converges to 1. []

+Uo E (7~(X( 1 -f,(z))) n--]

- b(X(1 -f,,{zll)) ,,/--1

3.2. Lemma.

× 1--1 /)()t(1 - f k ( z ) ) ) , (7) k=0 where U(1) = 1 was used in the derivation of (7). From (7) we have

n--1

(T(X(1--fk(x)))--D(X(1--fk(x))))

E

-fo(.,)))

k=l k-1

x I-I D ( X ( 1 - f r o ( x ) ) )

uo = f i / ) ( X ( 1

m=0

-£(0)))

n=0

uniformly converges if O < 1.

/ [ 1 - (7~(X) - D ( X ) ) Proof. Define g k ( x ) as

gk(x) = (7~(X (1

-fk(x)))-

b(X(1

oo

--fk(x))))

-

k-1

n--I

x I-I b ( a ( 1 m=0

--fm(X))).

- b(x(i -L(o)))) n-1

Since I1~(X(1 - A ( x ) ) ) l ~ 1 and f,,(x))) I ~< 1, it follows that

i g,<(x) I

1 - b(x(1

I/5(X(1 -

--fk(x)))

-fk(x))).

> 1 -- ad(1

x 1--[/)(X(1 - f k ( 0 ) ) ) k=0

]

.

(8)

3.3. Queue length distribution at an arbitrary instant

Next, we use the following inequalities, which were derived in the proof of Lemma 3.1:

/)(a(1

E (¢(x(a-/o(0)))

--fk(x)),

Let { p,} denote the stationary distribution of the number of customers in Q1 at an arbitrary instant. This probability distribution is defined as

and

p , = lira P r ( y ( t ) = i).

f k ( x ) > 1 - Ok .

This section derives the stationary distribution { Pi } using the rate conservation principle of the piecewise Markov process [13]. Let the mean interval between two successive regeneration points, say, t n and tn+l, be denoted by r - i . This can be calculated by

t ~

Then, I g k ( x ) I < X d ( 1 - f k ( x ) ) < X d o k. If p < l , Y~k oo= 1Ok converges. Thus, oo

oo

Igk(x)l < ~ Xdpk=Xdol( 1 -p). k=l

r -1 = ~ ui(ih + d ) + uoT,

k=l

Therefore, F~k=lgk( X ) uniformly converges.

[]

i=1

(9)

S. Sumita / Performanceanalysis of interprocessor communications

88

because the mean time between t. and tn+ 1 is equal tO ( i h + d ) when y ( t . - ) = i ( i > ~ l ) ; that equals T when y(t.-) = 0. It follows from (4) that

~ iu i = g ' ( 1 ) i=1

= ((1 - Uo)Xd+ uoXr)/(1 - p ) .

(10)

From (9) and (10), r-1 is given by r -1

=

( , o r + (1 - uo)d)/(1 - p).

(11)

(1) Mean waiting time in Q1. The mean waiting time in Q1, denoted by W1, can be calculated using the Little's formula: W a = La/X, (16) where L 1 denotes the mean queue length in Q1 at an arbitrarily chosen instant. Differentiating the generating function P(z) with respect to z and setting z = 1 with the help of L'Hrpital's rule, we have W, = P ' ( 1 ) / X

For simplicity of expression, we define v and v(2) as

Differentiating U(z) twice with respect to z and setting z = 1 with the help of L'Hrpital's rule, ~2

v = (1 - uo)d+ uoT, and v(2) = (1

u"(1) -

1 --0 2 [(1 - . o ) d 2 + Uor 2

u 0 ) d 2 + U0 T 2 ,

+ ( 2 p d + Xh(2))r-1].

respectively. Then, (11) can be rewritten as

--

Using the rate conservation principle of the piecewise Markov process, we have the following relationships between the stationary distributions { u i } and {pi}: oo

hpo + ru o = r ~_, u i,

(12)

i=0

hPi + rui = )~Pi-1 for i >/1.

(13)

Define the generating function P(z) as P ( z ) = oo i Ei=opiz. Then we have the following equation:

X po + ru o - rU( z ) X(1 - z )

(14)

Since (12) is equivalent to XPo = r ( 1 - u0), P(z) is finally given by

P(z)=

1 - U(z)

(18)

From (16), (17) and (18) r-' =v/(1 -p).

P(z) =

(17)

= g"(1)/[2•r-'].

W1

Thus P(z) has been expressed in terms of the generating function U(z) and other system parameters.

v(2)

2pd + ~h(2) .

v

(19)

1-p

(2) Mean waiting time in Q2. As described in Section 2, all the customers instantaneously move from Q1 to Q2 when the switch closes. Suppose that a customer finds ( n - 1) customers in Q1 when he arrives at the system. Then he occupies the n th position from the top of Q1. When that customer moves to Q2, he will occupy the (n - 1)th position from the top of Q2. This is because all the customers in Q1 move to Q2 in order of arrival. Thus the mean waiting time of this customer in Q2 equals ( n - 1)h. From this observation, the mean waiting time in Q2, denoted by W2, is given by

= E (n - 1)hp,_, = h P ' ( 1 ) n=l

(15)

~ r - l ( 1 - z) "

1 [ 2(1+0)

= OWl. (20) (3) Mean total delay. The mean total dealy W, including service time, is the sum of the mean waiting times in both queues and the service time. From (19) and (20), W is given by

l [ v (2) 2pd+~kh (2) ] -- + + h. v 1-7 J

3.4. Performance measures

W=

Performance measures, such as the mean waiting time and the mean queue length, can be derived using the generating functions U(z) and P(z). This section concerns the derivation of mean waiting time in each queue and mean total delay.

Here, note that

v = (1 - uo)d+ uoT and v(2)= (1 - u 0 ) d 2 + UoT2.

(21)

89

S. Sumita / Performance analysis of interprocessor communications

As noted in Section 2, the model analyzed in this section is identical to a single queue case of a cyclic service system with node dependent overhead. In the particular case of T = d, however, the model is a single queue system with node independent overhead and thus it is the same as a single queue, cyclic service system with gated service, the cyclic service system which was analyzed by Hashida [11]. From [11], the mean delay for the single queue case with gated service is given by 1 W=~

S (2)

s

+

- -- (2) 2 #s__+_aft ] / + h, 1-p J

20

15

W= ~ d+

20d + Xh (2)] ~C_~

+h.

/

[5 -4,

©

10

T=I!II/

5

I

I

.t

(23)

I

I

I

I

I

I

.2 .3 .4 .5 .6 .7 .8 .9 1.0 Traffic intensity p Fig. 3. M e a n total delay.

Fig. 4 shows for both gating figure it can be large, the mean

the mean waiting time in Q1, W], and clocked schemes. From this seen that when the offered load is waiting time in Q1 for the gating

10

3.5. Comparison with Clocked Scheme This section compares the gating scheme with the clocked scheme when the service time distribution is exponential with a mean of one. To compare both schemes, the following assumptions are made with regard to the clocked scheme. - The scan interval of the clocked scheme is equal to the scan interval of the gating scheme when no customers are found in Q1, i.e., in the clocked scheme, the switch closes at a constant interval of time T; - The overhead time of the clocked scheme is equal to that of the gating scheme. Fig. 3 shows the mean total delay as a function of offered load with a scan interval of T = 3, T = 5 and T = 10. In these results, the overhead time is fixed and equals 0.5. In the three cases above the mean total delay with the gating scheme is lower than that with the clocked scheme.

i

;"'" /l

(22)

If we set T = d in (21), the resulting equation agrees with (23). When (21) and (22) are compared, the first terms inside the brackets are different. For the system with node independent overhead, s and s (2} are given; for the system with node dependent overhead, v and v(2) are expressed in terms of u o and it is necessary to calculate u 0 by the application of (8).

/

',5r~

where s and s (2) are the mean and the second moment of switchover time distribution of the cyclic service system, respectively. In the context of this paper, s = d, and s (=) = d 2 and thus for the case of T = d we have

l[

Gating ..... Clocked h = 1; d = 0.5 H(x):Exponential

c~ ._c

Gating Clocked

9

.....

8

hH(x~ :Expd°ne=ntiOa~

7

t ,l°

cD

E

6

c~

5

"6

4

ca

3

,b

c-

/~

///

2 1

T=3 I

I

I

I

I

I

I

I

I

.1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 Traffic intensity p Fig.

4.

M e a n w a i t i n g time in QI.

90

S. Sumita / Performance analysis of interprocessor communications

15 .

.

.

.

Clocked

.

h= a •---

i f i

1;d=0.5

I r

H(x):Exponential

Oq

until it has finished processing all tasks in that processor; in the clocked scheme, a processor periodically requests message transfer to another processor. A queueing model has been presented for the performance evaluation of the gating scheme. Various performance measures, such as mean delay and mean waiting time, have been derived using the theory of piecewise Markov process. The gating and clocked schemes have been compared numerically in terms of the mean total delay and mean waiting time. It has been shown that the mean total delay for the gating scheme is lower than that for the clocked scheme.

Goring

i

10

t I

T--T10/ ",,' :D

:

T~5

t

i

/

j

',

Ii i

-.

5

Acknowledgment

0

t

I

I

I

I

I

I

I

.1

.2

.3

.4

.5

.6

.7

.8

I

.9 1.0

Traffic intensity p Fig. 5. Mean waiting time in Q2.

scheme is larger than that for the clocked scheme. In the gating scheme the server never doses the switch to get customers in Q1 until it has finished serving all the customers in Q2. Hence, the scan interval becomes larger as the offered load increases. Furthermore, it can be seen that the minimum value for W1 exists under some offered traffic condition. In the clocked scheme, the switch closes after every interval of time T. Thus the mean waiting time is independent of the offered load and equals T / 2 . T h e gating scheme is useful in shortening the mean total delay; the clocked scheme is useful in shortening the mean waiting time in Q1 even when the offered load is large. Fig. 5 shows the mean waiting time in Q2, W2, where W2 includes service time. This figure shows that W2 is lower than that with clocked scheme.

4. Conclusion This paper has proposed an algorithm, called the gating scheme, for message transfer among processors in an electronic switching system with distributed control. A major difference from the usual clocked scheme is that a processor never requests message transfer to another processor

The author would like to thank T. Katayama of N T T Communication Switching Laboratories for valuable discussions during the course of this work. He would also like to thank the anonymous referee for valuable comments, especially the indication of the similarity of the system considered in this paper to cyclic service systems.

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S. Sumita / Performance analysis of interprocessor communications

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