mixed parallel processors with interdependent blocking

APPLIED STOCHASTIC MODELS AND DATA ANALYSIS, VOL. 6, 85-100 (1990)

MIXED PARALLEL PROCESSORS WITH INTERDEPENDENT BLOCKING

NICO M. VAN DlJK Free University, Amsterdam, Postbus 7161, The Netherlands

SUMMARY

This paper studies a system of parallel service facilities or processors with mixed exponential and non- exponential queues, a non-exponential finite source input and interdependent arrival as well as departure blocking such as due to a common pool or shared resource. A concrete invariance condition upon the blocking protocol is provided. Under this condition the stationary busy source distribution is shown to be of an insensitive product form. The result unifies and extends known product form results as will be illustrated by several examples. K E Y WORDS Parallel processors Finite source input Blocking Invariance condition

Product form Insensitivity

INTRODUCTION

Applications of queueing networks, such as in telecommunications, computer performance evaluation and manufacturing, frequently feature situations in which a number of parallel servers have to share some common pool or buffer. For instance, in communication networks calls from various subdirections are often transferred via central transmitters with finite capacities (e.g. satellites). In voice-data communication along digitized channels voice and data have separate reserved time slots (to be seen as servers) but cope with limited common storages or transmission channels. Store and forward buffers in computer networks usually compete for joint finite output devices. In parallel programming different program modules may have to be handled by separate processors attached to the same drum. Manufacturing usually involves different parts that are to be transported, served and stored jointly. Also, a manufacturing material handling system (MHS) is often centralized to a single processor with limited storage and service capacity.

The literature on parallel processors is restricted to Poissonian arrivals, exponential services and total capacity constraints. 1-6 Under these assumptions product type formulae have been established. Unfortunately, however, exponential services are often far from reality. Moreover, in various practical situations inputs are generated by finite non-exponential sources rather than by external Poisson streams.

Further, other blocking phenomena rather than total capacity limitations can be in order such as randomized accessing, class priorities and blocked departures.

This paper aims to provide the extension of parallel processor systems to

1. a non-exponential finite source input 2. combined exponential and non-exponential processors 3. more complex processor or source interdependent blocking.

8755-OO24/90/02OO85- 16$08 .OO 01990 by John Wiley & Sons, Ltd.

Received 6 October 1988 Final revision 27 November 1989

86 N . M . VAN DlJK

The main results are:

(i) a product form and partially insensitive expression (ii) a concrete blocking invariance condition

(iii) a self-contained straightforward proof (iv) a number of examples with novel aspects of

(a) a non-exponential finite source input (b) job-class dependent FCFS-queues (c) non-coordinate convex blocking (d) non-exponential services.

Product forms and their relationship with notions of partial balance have been extensively studied over the last decade. '- l 3 For exponential queueing networks without blocking phenomena a product form is a common feature. For systems with blocking, product forms are much more restricted. l o - 1 2 * 1 4 * 1 5 Further, the extension to non-exponential systems has generally been shown only in abstract frameworks without explicit blocking phenomena.13q16-'9 Moreover, the application of these results to concrete models still requires technical details and appropriate transformations.

The scenario is as follows. First, in section 2 we present the model, the essential blocking condition and several examples. Next, in section 3 the general product form result is derived.

2 . MODEL

The system under investigation contains a finite number of sources, say numbered 1, ..., M , and a finite number of parallel service facilities, say numbered 1, ..., P. A source requires services from one and the same service facility. A source is said to be of type t if this is service facility f . Let th be the type-number of source h and Mt the number of type-t sources. Hence, M I + A42 + ... + Mp = M .

Idle-busy and blocking mechanism

Each of the sources alternates between idle and busy periods as follows. After completing a so-called scheduling time during which it is called idle, a source h of type t requests a service by facility f. If upon this request other sources H = ( h l , ..., h,) are already busy, the request is accepted with probability A (h I H ) , and the source becomes busy up to service completion. With probability 1 - A (h I H ) it has to remain idle and to restart a new scheduling time instantaneously. Conversely, when a busy source h completes a service while the other sources H = (hl , ..., h,) are still busy, source h will become idle with probability D(h I H ) , and start a new scheduling time. With probability 1 - D(h I H ) it has to remain busy and to receive a new service by facility t as a newly accepted service request.

Remarks I 1 . Many practical examples will involve only the function A (. I .) while D(. 1 .) = 1. The

inclusion of the function D(. I .), however, is completely dual and allows us to model service delays or priorities (cf. examples 5 and 6).

2. The assumption of having to repeat a new scheduling or service time upon blocking is common for communication systems. 1 - 3 7 2 0 9 2 1 It effectively comes down to interrupting or stopping a scheduling or service time when the source status is currently not allowed to change.

MIXED PARALLEL PROCESSORS 87

Blocking invariance condition

We need to impose a condition upon the blocking functions A ( . 1 .) and D(. 1 .) in order to guarantee a product form result. To this end, let a state H = (h l , ..., h n ) denote that sources h1, ..., h, are busy, where hl , ..., h, are given in increasing order. The ordering is given merely for presentational convenience in the condition below and does not play any role itself. Let state 0 denote that all sources are idle. Let X b e a set of states which contains 0 and such that out of any state of this set any other state in this set can be reached by some sequence of possible scheduling time and service completions (that is, with positive corresponding A (. 1 .) and D(. 1 .) values), while states outside this set cannot be reached in this way out of any state from this set. (Under exponential scheduling and service time assumptions the set Xwould be called the irreducible set in Markov chain analysis.) Further, for a state H = (hl , ..., h,) of busy sources, let H + h or H - h denote the same state with source h added (+) or deleted ( - ) as a busy source.

Condition I For any H = ( h l , ..., h,) E .W, some value P(H) and some I < n:

while

for all i = 1, ..., n and

for all permutations (i,, ..., in) E (1, ..., n ) for which the denominator is positive. Inequality (1) guarantees that the product in (3) has a positive denominator for at least one

permutation, and (2) guarantees that if the denominator of this product is zero then the numerator is also equal to zero, so that the product can be chosen equal to P ( H ) .

Remarks 2 1 . Quite frequently, condition 1 can be particularized to a form which depends only on the

numbers of jobs at the various facilities. 2. Equation (3) can be seen as a specified form of the well-known Kolmogorov criterion"

for a Markov chain to be reversible. The present applied form, however, has not been reported.

Blocking examples

Below we will present several examples of blocking structures that satisfy condition 1. The 'coordinate convex examples' have each been studied in the literature individually. The other examples seem to be new.

(i) Coordinate convex blocking Assume that D ( . .) = 1 while

A (h 1 H ) = 1 (H + h E C ) (4)

88 N. M . VAN DIJK

where C is some set of states such that for all j :

H € C*H - hj € C ( 5 )

Roughly speaking that is, transitions that will keep the state within C are not blocked at all while transitions that would bring the state outside C are completely blocked. As will be illustrated by examples 1-3 below, such a blocking mechanism is quite natural when merely finite capacity limitations or strict source exclusions are in order. In correspondence with the l i t e r a t ~ r e ” ~ we call this type of blocking ‘coordinate convex’. Note that for any coordinate convex blocking:

P ( H ) = 1 (6)

Example 1 (multi-storage allocation). 3,22923 Suppose that the sources are classified into P classes. A busy source of class p occupies bp storage places simultaneously (see Figure 1). With B the total number of storage places and np the number of busy class-p sources, C is the set of states H satisfying

P

Figure 1 Figure 2

Example 2 (circuit switching). 24 Consider a transmission system with four different source classes and the end-to-end configuration shown in Figure 2.

The number Nj represents the number of trunks in trunk-group j . A request from a class-i source is accepted only when all trunk groups along its end-to-end path have at least one available trunk. A busy source of class i uses one trunk from each of the trunk groups along its path. With np the number of busy class-p sources, C is the set of all states H such that

ni < Ni, i = 1, ..., 4

nl + n2 < N5 n3 + n4 < N6

nl + n2 + n3 + n4 < N7

Example 3 (carrier sense multiple accessing). 25*26 Let a source represent a transmitter/receiver node that communicates with neighbouring nodes. The nodes can be graphically presented such that adjacent nodes cannot transmit at the same time. Physically this


Figure 3

is realized by so-called ‘carrier sensing’ before a transmission starts. For example, in Figure 3 a transmission of node 5 prohibits all other nodes to transmit, but a simultaneous transmission of nodes 1 and 4 is possible.

By considering a transmission as a service and the time up to a next transmission request as a scheduling time, the above framework applies with a separate service facility reserved for each source. With N ( h ) the set of neighbours from node h, the communication restrictions are modelled by the coordinate convex blocking parametrized by

C = (N I H has no neighbours ]

A (h 1 H ) = I(h 4 N(hi) for all i = 1, ..., n)

(ii) Random accessing In the examples so far, the function A (. I .) only has values 0 and 1. Access probabilities other

than 0 and 1, however, appear most naturally. For example, in Erlang’s classical ideal grading a fixed number of servers, say K, was randomly selected to find a free server. With N the total number of servers, of which n are currently busy, a service request is then accepted with probability

Clearly, generalizations to class-dependent gradings can be provided. An example of more present-day interest is the following.

Example 4 (access collision detection). Again consider a communication system with sources representing transmitters. For simplicity assume that all sources can transmit simultaneously and have the same exponential distribution with mean y for the time up to a transmission request. However, owing to discrete-time slotting simultaneous transmission requests may occur and collisions at some common input device can arise (e.g. before a transmission can start some address is to be retrieved). In that case requests are aborted. With d the length of a time-slot, this feature is standardly modelled by

(8) A (h I H ) = - f 4 M - n - I V Y

Assuming D(. I .) = 1 the invariance condition (1) is satisfied with

P ( H ) = - dn(2M - n - 1)/(2r) (9)

(iii) Interdependent servicing Service by the facilities may be obtained from a common resource, so that the service speeds

of the different source classes and parallel facilities are interdependent. This can be modelled by use of the function D ( . I .). Particularly, allowing D ( . I .) to be 0, servicing of a facility may be completely stopped provided also that arrivals are appropriately controlled.

90 N . M . VAN DIJK

Example 5 (service delay). Assume that all facilities are delayed by a factor 2 when the total number of jobs exceeds some threshold L . This is modelled by

With A (. I .) = I , (3) holds with

Example 6 (service interruption). Consider two source-types 1 and 2 which are separately served by facilities 1 and 2, respectively. As type-1 sources have a higher priority, service requests from type-2 sources are rejected and servicing at facility 2 is stopped (interrupted) as soon as the number of busy type-1 sources exceeds some threshold L (see Figure 4).

With t!, the type-number of source h and nl the number of busy type-I sources in state H , the corresponding parametrization is

The invariance condition (1) is satisfied here with

P ( H ) = 1

Figure 4

Service mechanism

The way in which busy sources of type t receive service from facility f is governed by positions and a discipline parametrized by a tuple (ft, pt, 1 3 ~ ) as follows. When n sources of type t are busy, then

(i) J ( n ) is the total rate at which service is provided (service capacity) (ii) p r ( p I n ) is the fraction of this capacity assigned to position p , p = 1 , ..., n

(iii) 6, (p I n ) is the probability that a source of type t which becomes busy is assigned position p , p = 1 ,..., n + 1.

When a source at position p completes its service, the sources previously at positions p + 1 , ..., x are shifted to the positions p , ..., x - 1. When a source which becomes busy is assigned position p , the sources previously at positions p , ..., x are shifted to the positions p + 1, ..., x + 1. Further, the natural assumption is made that f t ( x ) > 0 for x > 0 and Cppt(p 1 x ) = C,61(p I x) = 1. The following distinction will be essential regarding exponentiality conditions to be imposed.

A facility t is said to be of type 1 (non-symmetric) if it adopts the above description without


further condition. A facility is said to be of type 2 (symmetric) when it also satisfies

d P / n ) = 6 r ( P ( n - I ) , p = I , ..., n ( n > O ) (13) In accordance with Kelly, " a facility of type 2 is called symmetric. Referring to this reference for details, various service disciplines can be parametrized in the above manner, of which the most notable are the standard four BCMP disciplines

FCFS (type 1): (first-come-first-served) PS-I (type 2): PS-cr, (type 2 ) : LCFS-pre (type 2 ) : (last-come-first-served pre-emptive).

(processor sharing-single server) (processor sharing-infinite server)

Let .'/'denote the set of all symmetric facilities (i.e. of type 2 )

Scheduling and service times

Let Ah and Bh be the scheduling time and service requirement distribution function respectively of source h. Without loss of generality, assume that these distributions are absolute continuous with respective density functions a h ( . ) and o h ( . ) and means yh and 7 h .

Partial exponentiality assumption

For each non-symmetric facility t $9, all sources h of type t and some parameter pI:

That is, the service at a non-symmetric facility is required to be exponential with one and the same parameter for all sources associated with this facility. Hence pf = ( 7 h ) - ' for all h with t / [ = t 4 .y:

3 . PRODUCT-FORM RESULT

This section shows that the invariance condition 1 will be responsible for a partially insensitive product-form expression. Theorem 1 below is the technical key theorem. Theorem 2 is the more practical corollary which deals only with the idle and busy status of a source.

As different sources, service facilities, positions and service times are involved, some notational complexity is unavoidable. We denote by

[ S , P , RI = ((Sl, PI, rz), ..., ( S M , PM, rM))

for each source h:

(i) s h : the current status with Sh = 0 for idle and Sh = 1 for busy (ii) PI,: the current position-number at facility t h where t is the fixed type number of source

h, when source h is busy (Sh = I) , whereas this number is artificially fixed at ph = 0 when source h is idle (sh = 0)

(iii) r h : the residual scheduling time when the source is idle (s /~ = 0) and the residual service requirement when the source is busy (Sh = 1).

Furthermore, for a given state of the above form, denote by H the corresponding busy source vector and by ri = (nl, ..., np) the numbers of busy sources of type f , t = 1, ..., P . Finally, recall that each source h has a type number t h which is fixed once and for all, h = 1 , ..., M .

92 N . M . VAN DlJK

Without loss of generality, also see remark 3 below, assume that there exists a unique stationary distribution with densities denoted by a( [ S , P, R ] ), with H , the corresponding set of busy sources, restricted to the set Ye. Without further mention, the product-form expressions that will be used below are always assumed with H restricted to this set .X.

Theorem 1

Under the invariance condition 1 and with c a normalizing constant, we have

Pro0 f We need to verify the global balance or equivalently stationary forward Kolmogorov

equations under the standard assumption that these have a unique solution. For notational convenience, for a given state [ S , P, R ] and source h, let

[s, p , R1 h + (s, p , r ) h = [(SI, PI, rl), ..a, ( s h - 1, P h - 1, r h - 11 , (sl P , h) , ( s h + I , p h + 1, r h + 1 )> .. ., ( S M , P M M , r M ) I

In words that is, the state with for all sources j # h the specifications ( S j , P j , r j ) as in [S , P, R ] but with for source h the specification (s, p , r ) of status s, position p and residual scheduling or service requirement r. Further, for a given function f ( x ) we denote by f ( 0 ' ) the right-hand limit at x = 0, i.e. f(0' ) = limx10 f ( x ) .

Now under assumption that the system is stationary, the forward Kolmogorov equations require that in any state [ S , P , R ] the 'total rate of change' should be equal to 0. (Formally, the rates involved are derived in the standard manner of considering a time-point t + A t , equating the possible inflows from time-point t , dividing by A t and letting A t tend to 0.) When the system is in state [S , P , R] with busy source configuration H of which nr busy sources are of type t , for each source h, with specification ( S h , P h , T h ) , the following rates are hereby involved:

(a) Zdle source. When the source is idle ( s h = 0):

as due to the reduction of residual scheduling time by one unit per unit of time,

a( [ S , P, R1 h -k ( 1 9 090' ) h ) 1 - A 1 a h ( r h )

as representing that the source has just been prior to completion of a scheduling time and that upon completion of this scheduling time it was blocked so that a new scheduling time of duration r h had to be selected with density a h ( n ) , and

3 R ( [ S , p, R1 h + (2% P , 0' ) h ) f r ( n r + I)'f't(p I nr)D(h I H b h ( r h ) p = I

reflecting that the source has just completed a service at some position p at the facility of its source type t = fh where it was provided service at a ratefr(nt + l)cpI(p 1 n, + l ) , upon which with probability D(h 1 H ) it became idle and upon which it selected a scheduling time of duration r h with density CYh(r/ t ) .


(b) Busy source. When the source is busy (sh = I), say at facility t = t h with position pk:

a - a([& p, f?I ) f t ( n f ) P f ( p I nr) arh

as due to the reduction in residual service requirement by the service capacity f t ( n r ) p t ( p f l 1 n f ) provided per unit of time,

n , + 1

p = 1 .( [s, P, Rl h + (2, P , 0' )h)fr(nf)pt(Ph 1 nr) [ 1 - D(h 1 ff - h)] &(Ph 1 ni - 1)Ph(rh)

as representing that the source has just been prior to completion of a service at some (other) position p at this same facility t , where it was provided service at a rate f t ( n f ) p t ( p I n f ) , and that upon completion of this service it was blocked to become idle with probability D(h 1 H - h ) so that a new position p h at this facility t and service requirement rh were selected with probability &(ph I nf - 1) and density Ph(rh), respectively, and

reflecting that the source has just completed a scheduling time upon which it became busy with probability A (h 1 H - h ) and selected a position ph at facility t = t h and service requirement r h

with probability &(ph I nt - 1) and density Ph(rh), respectively. By summing all these rates of change for all idle (sh = 0) and busy (sh = 1) sources and

collecting for any facility t the terms for the busy sources of type t h = t , the forward Kolmogorov equations become

r h ) = o I Assuming that (16) has a unique probability density solution T(. ) , it thus suffices to verify (16) with (15) substituted for T( . ) . This will be achieved by separately investigating the terms involved for an idle source h under (i), for a busy source h at a symmetric facility under (ii) and for all busy sources at a specified non-symmetric facility t under (iii).

( i ) h fixed, sh = 0. Consider a fixed source h with Sh = 0. Then by (15)

a - a([& P,R])= - arh

As D(h 1 H ) = 0 implies [ 1 - A (h 1 H)] = 1 by virtue of (2), the term between I . f for fixed h with sh = 1 is thus equal to 0 when D ( h ( H ) = O . When D ( h l H ) > 0, for arbitrary

94 N . M . VAN DIJK

p = 1 , ..., n, + 1 we conclude from (15) and the invariance condition 1:

T ( [ S , P, R l h f (2 , p , o + ) h ) = T ( [ S , P , R l h -k (liOio+)h) ::i 1;; .hb(nfh -k 1) (18)

By substituting (17) and (18) and using the identity 2;~'i' p r ( p I nt + 1) = 1, this proves that the term between ( . ) for fixed h with s h = 0 is also equal to 0 when D(h I H ) > 0.

(ii) t E Yfixed, h fixed with Sh = 1 and t h = t . Now consider an arbitrary fixed t E Y and source h with Sh = 1 and t h = t , that is with a service requirement at a symmetric queue. Then by (15) we have for arbitrary p :

As A ( h 1 H - h ) = 0 implies [ I - D(h 1 H - h ) ] = 1 by virtue of (2), the term between braces: ( . ) , for fixed h with s h = 1 is thus equal to 0 when A ( h I H - h ) = 0. When A ( h 1 H - h ) > 0 , for arbitrary p = 1, . . . , n, + 1 we conclude from (1 5) and the invariance condition 1 :

By substituting (19) and (20), using the identity C;LI pr(p I nr + 1) = 1, and recalling the symmetry property (13) for a symmetric queue t , this proves that the term between braces: ( . ) for fixed h with s h = 1 and t h E ,Yis also equal to 0 when A (h I H - h ) > 0.

(iii) t ( .Yfixed. Finally, consider an arbitrary fixed non-symmetric queue t $ .Y' and note that by assumption for all h with t h = t:

O h ( r h ) = e-"""

[l - & ( r h ) ] = e-Prrlt

As a consequence, for any h with s h = 1 and t h = t , we can conclude from (15):

a - T ( [ S , P, RI ) = - P r T ( [S, P, RI 1 a r h

T ( [S, P, R] h + ( 2 , p , O + ) h ) = e+pirhT( [ S , P, R] )

with the latter equality provided A (h 1 H - h ) > 0. For A (h I H - h ) = 0, however, we have [ 1 - D(h I H - h ) ] = 1. By substituting these relations in (16) for fixed t , the sum over all h with

Sh = 1 and t h = t in either case thus reduces to

P z f t h ) * ( [ S , P7 RI ) - c Pr(P I nr) + c St(p I nr - 1) = 0 t P P 1

As the global balance equation (16) is hereby verified, the proof is completed. 0

Let the states [ S , P] = ((sl, PI), ..., ( S M , p ~ ) ) and [SI = (SI, ..., S M ) denote the status Sh and position P h of each source h . The corresponding stationary probabilities are presented by a([S, PI ) and T ( [ S ] ) with H restricted to the set <%.


Theorem 2

With c and 5. = c(yl ... YM) normalizing constants, we have

I 11. \ - I

Pro0 f

olr recalling the analytical identity Expression (22) is directly obtained from (15) by integrating over all possible residual times

for arbitrary positive functions f~(.), ...,f~ (.), and noting that m

11 - Ah(r)1 d r = y h 0

Expression (22 ) follows from (21) by substituting c = i/ [yl ... Y M ] and noting that for given values n,, t = 1, ..., T, there are ( n l ! ) ... ( n T ! ) different states of the form [ S , PI with specified positions for the sources in service for which [ S ] only specifying the idle or busy status of the sources is fixed. 0

Remark 3 (general distributions) The assumption of absolute continuous think and service distributions is essentially used in

the above Markovian proof technique. It is well known, however, that such distributions approximate arbitrary distributions arbitrarily closely, in the sense of weak convergence, such as by using mixtures of Erlang distributions. '' By standard though highly technical weak convergence limit theorems for sample paths on so-called D-spaces, 16,27$28 Theorem 2 can therefore be extended to general distributions. For the technical details we refer to these references.

Remark 4 (normalizing constant c) Clearly the actual computation of the product forms presented still requires the normalizing

constant c to be determined. This can be most complex. To this end techniques developed in the literature such as in References 20, 29 and 30 can be exploited.

4. APPLICATIONS

This section will briefly examine some of the blocking examples of section 2 in more detail so as to illustrate the product form result and its insensitivity more concretely.

96 N. M . VAN DIJK

4.1. Multi-storage allocation; example 1

Reconsider example 1 with M, sources of class p = 1, ..., P and for each source class 2 a separate service facility t with service characteristics (h, 6[, pr) (see Figure 5). These facilities though are subject to a common storage constraint of size B where a source of type t requires br places simultaneously, as reflected by (7).

Sources of type I have a general scheduling time with mean yr. The service requirement is also allowed to be generally distributed with mean T[ provided that facility t is symmetric (i.e. satisfying (13)) but has to be exponential with parameter 117, when facility t is non-symmetric.

Let n = (n l , ..., n,) denote the number nt of busy sources of class t and note that (7) directly guarantees ( 5 ) and thus also condition ( 1 ) with P ( H ) = 1. Hence, by (6 ) , (8) and simple combinatorics:

where c is a normalizing constant, represents the stationary distribution. Particularly, with all facilities corresponding to an infinite server queue all service requirements can be non- exponential and we obtain the extension of Engset’s classical formula:

B

Figure 5

4.2. Carrier sense multiple accessing

( i ) Standard CSMA; example 3 Reconsider example 3 with A4 transmitters and for each transmitter h a given set of

neighbouring transmitters N ( h ) . (see Figures 3 and 6). Upon transmission request of a node, a new transmission by that node is to be rescheduled if one of the neighbouring nodes is already transmitting. The scheduling and transmission times are allowed to be generally distributed with mean yh and T h , respectively, for node h .

With C the set of busy node configurations H which excludes two or more busy neighbours at the same time, condition ( 5 ) is directly verified.

Further, to apply the framework of section 2 , visualize all ongoing transmissions as services that take place at one and the same infinite server facility t = 1 . As such a facility is symmetric with f r ( k ) = k , we obtain from (6 ) and (22):


(ii) Rude-CSMA 26

As an extension of t h above standard CSMA model which is pro x e d in Reference 26, assume that a request by a node h to start a transmission while other nodes H are currently transmitting is accepted with probability

A ( h l H ) = x N " ( h ) y , V ' ( h )

where x and y are fixed given values and where N o ( h ) is the number of idle neighbours of h and N ' ( h ) is the number of busy neighbours of h .

For example, with x = 1 and y = 0 one obtains the model of (i). As per remark 2.2, one can conclude that condition (3) is satisfied provided one can find a function P ( H ) such that

P ( H + h ) = A ( h I H ) P ( H )

This in turn is easily verified for p ( ~ ) = x B " ( H ) B ' ( H ) Y

where B o ( H ) is the number of pairs of idle neighbours in state H a n d B ' ( H ) is the number of pairs of busy neighbours in state H .

As under (i), the result from Reference 26 is now directly concluded as

( i i i ) Collision detection; example 4

will lead to In a similar manner, the communication system of example 4, in which collisions can arise,

h t H

regardless of service distributional forms provided that all busy sources can transmit at the same time, whereas

a ( ~ ) = n! Idyl (29)

when only one busy source can transmit at a time (such as in a single bus Ethernet system) and under the assumption that all sources have exponential transmission times with the same parameter 1/7. (Note that the scheduling times are here still allowed to be non-exponential.)

e - d n ( 2 M - n - l ) ( 2 - r ) '

4.3. Service delay

(i) Common service delay; example 5 Reconsider examples with single-server facilities which are delayed by a factor 2 as soon as

98 N . M . VAN DlJK

the total number of busy sources exceeds the threshold L . The scheduling times can be generally distributed but the service requirements are all exponential with respective means yt and Tf for sources of type t . Then by virtue of (lo), ( l l ) , (22) and some combinatorics we obtain;

(ii) Service interruption; example 6

as in (i), we directly conclude from (12) and (22) For the two-facility example 6 with scheduling time and service requirement characteristics

(31) n ( i ) = c[71/ylI "1[72/y21 "'[(Mz - nz)!] - I

5. EVALUATION

A framework is presented of mixed exponential and non-exponential parallel stochastic service facilities which serve a finite number of sources in an interdependent manner. A concrete condition in terms of the interferences is provided in order to conclude an explicit product-form expression for the busy source distribution. Coordinate convex interferences such as reflecting synchronous servicing, common finite service resources, circuit switching or CSMA communication structures contain a simple but broad class of examples. But also examples with service interruptions or source-class priorities and randomized blocking such as in rude CSMA are covered here. The results are of interest also for inclusion in interconnected communication networks or distributed systems.

ACKNOWLEDGEMENT

The comments of the referees which helped to clarify the presentation are appreciated.

GLOSSARY OF SYMBOLS

probability that a service request by source h is accepted scheduling time distribution for source h service requirement distribution for source h normalizing constants (see (15), (21), (22)) coordinate convex region, i.e. satisfying ( 5 ) probability that a source h which completes service becomes idle service capacity at facility t with nr busy sources of type t a particular source set of current busy sources set of admissible configurations H number of sources number of busy type-t sources current position at a facility of source h number of parallel facilities as defined by (3)


rh

sh

[ S , P, RI = ((st, PI, rl), . . ., ( S M , PM, T M ) )

[ S , PI = (61, PI ), . . . , ( S M , p ~ ) ) [Sl = (SI, ..., SM) $1 t

ti,

f f h ( . )

o h ( * )

Y h

SI(P I n ) pr = T i 1

PAP I n l ) 7 h

current residual scheduling time or service requirement of source h current status of source (Sh = 1 means up and s h = 0 means down)

specification of status (idle or busy), position and residual time for all sources h

similar restricted to status and positions similar restricted to status only

set of all symmetric facilities (see (13)) number of a service facility type (facility) number of source h scheduling time density for source h service requirement density for source h mean scheduling time for source h probability position p is assigned to new busy source exponential service parameter for t h = t ( 9’ fraction of this capacity assigned to position p mean service requirement for source h

REFERENCES

I . G . J . Foschini and B. Gopinath. ‘Sharing memory optimally’, IEEE Trans. Comrn., 31, 352-359 (1983). 2 . F . Kamoun, and L. Kleinrock, ‘Analysis of shared finite storage in a computer network node environment under

general traffic conditions’, IEEE Trans. Comm. Corn-28, 992-1003 (1980). 3 . J . S . Kaufman, ‘Blocking in a shared resource environment’, IEEE Trans. Comm., 29, 1474-1481 (1981). 4 . S S . Lam and M. Reiser, ‘Congestion control of store and forward networks by input buffer limits’, IEEE

Trans. Comm., Corn-27, 127-133 (1979). 5. J , Swiderski, ‘Unified analysis of local flow control mechanisms in message-switched networks’, IEEE Trans.

Comrn. , Corn-32, 1286- 1293 (1984). 6. A . K . Thareja and A. K. Agrawala, ‘On the design of optimal policy for sharing finite buffers’, IEEE Trans.

Comm., Corn-32, 737-740 (1984). 7 . F . Baskett, M. Chandy, R . Muntz and J . Palacois, ‘Open, closed and mixed networks of queues with different

classes of customers’, JACM, 22, 248-260 (1975). 8 . K . M . Chandy, J . H . Howard and D. F . Towsley, ‘Product form and local balance in queueing networks’,

9 . K . M . Chandy and J . Martin, ‘A characterization of product form queueing networks’, JACM, 30, 286-299 (1983).

10. A . Hordijk and N. M. Van Dijk, ‘Networks of queues, Part 1: Job-local-balance and the adjoint process; Part 11: General routing and service characteristics’, Lecture Notes in Control and Information Sciences, 60, Springer Verlag, 1983. pp. 158-205.

1 1 . F . P . Kelly, Reversibility and Stochastic Networks, Wiley, 1979. 1 2 . .I. Walrand, Introduction to Queueing Networks, Prentice-Hall, 1987. 1 3 . P . Whittle, ‘Partial balance and insensitivity’, J . Appl. frob., 22, 168-176 (1985). 14. A . Hordijk and N. M . Van Dijk, ‘Networks of queues with blocking’, Performance 1981, North-Holland, 1981,

1 5 . B. Pittel, ‘Closed exponential networks of queues with saturation. The Jackson-type stationary distribution and its asymptotic analysis’, Math. Oper. Res., 4, 357-378 (1979).

16. A . D. Barbour, ‘Networks of queues and the method of stages’, Adv. Appl. f rob . , 8, 584-591 (1976). 17. D . Y . Burman, ‘Insensitivity in queueing systems’, Adv. Appl. frob., 13, 846-859 (1981). 18. A . Hordijk and N. M. Van Dijk, ‘Adjoint processes, job-local-balance and insensitivity for stochastic networkr’,

Bull. 44th Session In!. Srat. Insr., 50, 776-788 (1983). 19, R , Schassberger, ‘The insensitivity of stationary probabilities in networks of queues’, Adv. Apl. f rob . , 10,

20. I . F . Akyildiz and H . Von Brand, ‘Computation of performance measures for queueing networks with reversible routing and rejection blocking’, to appear in ACTA Informatica, Springer Verlag .

21. S. S . t a m , ‘Store and forward buffer requirements in a packet switching network’, IEEE Trans. Comm., Com-

22. L . Green, ‘A queueing system in which customers require a random number of servers’, Oper. Res., 28,

JACM, 24, 250-263 (1977).

pp. 51-65.

906-912 (1978).

24, 394-403 (1976).

1335 - I346 (1980).

100 N. M. VAN DIJK

23. W . Whitt, ‘Blocking when service is required from several facilities simultaneously’, AJ&J Technical Journal,

24. D . Y. Burman, J . P . Lehoczky and Y. Lim, ‘Insensitivity of blocking probabilities in a circuit switching network’, J . Appl. f r o b . , 21, 850-859 (1984).

25. R. R. Boorstyn, A . Kershenbaum, B. Maglaris and V. Sahin, ‘Throughput analysis in multihop GSMA packet radio networks’, IEEE Trans. C o m m . , 35, 267-274 (1987).

26. R . Nelson and L . Kleinrock, ‘Rude-CSMA: a multihop channel access protocol’, IEEE Trans. C o m m . , Corn-33,

27. A . Hordijk, and R . Schassberger, ‘Weak convergence of generalized semi-Markov processes’, Stochastic Processes & A p p l . , 12, 271-291 (1982).

28. W . Whitt, ‘Continuity of generalized semi-Markov processes’, Math. Oper. Res., 5 , 494-501, (1980). 29. D . Mitra and J . McKenna ‘Asymptotic expansions for closed Markovian networks with state-dependent service

rates’, JACM, 33, 568-592 (1988). 30. E . Pinsky and Y . Yemini, ‘A statical mechanics of some interconnection networks’, Performance ’84, North-

Holland. 1984.

64, (8), 1807- 1856 (1985).

785-791 (1985).

mixed parallel processors with interdependent blocking

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