Product forms for random access schemes

Computer Networks and ISDN Systems 22 (1991) 303-317 North-Holland

303

Product forms for random access schemes N i c o M. v a n D i j k Free University, Faculty of Economics and Econometrics, De Boelelaan 1105, 1081 H V Amsterdam, The Netherlands

Abstract Van Dijk, N.M., Product forms for random access schemes, Computer Networks and ISDN Systems 22 (1991) 303-317. Abstract A unifying framework is presented to conclude explicit product form expressions for the steady state distribution of busy sources (transmitters, lines) for random access communication protocols. Transmission times and packet lengths are generally distributed. The main results are: (1) An insensitive product form expression. (2) A concrete condition in terms of system protocols. (3) A generalization of product form random access protocols. Here, insensitivity means that the underlying random distributions (transmission times and packet lengths) play a role through only their means while a product form stands for factorization in individual transmitters or links. These results unify and extend known results. Particularly its includes CSMA-protocols with (i) Non-exponential transmissions and packets. (ii) State dependent transmission speeds. (iii) Link selective characteristics. A variety of "novel" examples is given such as with hierarchical circuit switching, synchronous serving, randomized grading, message priorities, error probabilities, link selective transmissions and an extension of rude-CSMA. Keywords: product form, broadcasting, A L O H A / B T M A / C S M A , coordinate convex interference, insensitivity, random access schemes, invariance condition, packet/circuit switching.

1. Introduction

Various packet or circuit switching random access schemes for computer, broadcasting or telecommunication networks have been introduced and investigated over the last decades (cf. [3,15,19-21,23,27-31,39]). Most notably are the ALOHA (e.g. [19,27,31]) and CSMA (e.g. [19,20, 27,31]) packet switching protocols and their various extensions (e.g. [3,23,30,31]). Explicit product form expressions for the steady state busy source distribution for CSMA-protocols were established under the assumptions of exponential packet lengths and transmission times and simple interactions such as arising in single-hop radio packet networks. These results were extended to multihop interactions which take into account the so-called "hidden terminal problem" in [23] and [3]; in [23] to the so-called "rude-CSMA" protocol and in [3] to link-selective messages and non-exponential packet lengths. A relaxation of the results in [23] to transmitter dependent parameters was given in [9]. The results from [3] were generalized to non-

exponential packet lengths and transmission times as well as more general system protocols in [4-6]. Most notably, in these latter references both sufficient and necessary product form conditions were provided for multihop packet-radio networks with nonrandomized access protocols. This paper provides a simple framework which unifies and extends the above CSMA product form results. Particularly, it allows: (i) Non-exponential transmissions and packets. (ii) Randomized random access mechanisms. (iii) State dependent transmission speeds. Generally, the main results are: (1) An insensitive product form steady state distribution. (2) A product form condition in concrete terms of system protocols. (3) A generalization of product form random access protocols. Here, insensitivity means that the steady state busy source distribution depends on the transmission times and packet lengths only through their

0169-7552/91/$03.50 © 1991 - Elsevier Science Publishers B.V. All rights reserved

304

N.M. van Dijk / Productforms for random access schemes

means. A product form stands for factorization to individual components or stations. This product form result is related to product form results in the extensive literature on queueing networks (cf. [7,8,12,13,17,37]). However, to make the proper transformations and to verify the abstract conditions in these references for the present application, essentially the self-contained proof that is given in this paper would be required. Moreover, the results of these references would not directly lead to sufficient conditions in terms of concrete random access protocols. In the particular framework of multihop packet-radio networks, necessary and sufficient conditions for product form results have been derived in [4-6]. In the general setting of r a n d o m access schemes, however, no sufficient concrete conditions in terms of system protocols seem to be available. To this end, a general invariance condition will be provided. It so turns out that various known product form telecommunication examples can be unified (e.g. examples 2.1-2.4). But also new product form transmission examples (e.g. examples 3.2-3.4) and (see Section 5) a generalization of the rude-CSMA protocol of [23] as well as the link-selective results in [3] are easily concluded. The organization is as follows. First, in Section 2 the model is presented. Next, in Section 3 the essential product form condition is given and illustrated by several examples. The product form result is established in Section 4. Finally, the particular models of [23] (Rude-CSMA) and [3] are discussed as special examples. An evaluation concludes the paper.

2. M o d e l

Below we first give an abstract model formulation so as to avoid lower-level technicalities at this point and to allow various interpretations later on as will be illustrated by examples. Abstract formulation

Consider a system of N nodes (e.g. representing transmitters), numbered 1 . . . . . N. Each node is alternatively in a busy (e.g. transmitting) and idle (e.g. non-transmitting) mode determined as follows. When a node h becomes idle, it requires a r a n d o m amount of "idle-service" (e.g. time required to schedule a next transmission) with distribution function I h. U p o n completion of an idle service it requests to become busy. If upon this request other nodes h I . . . . . h n are already busy, with probability A ( h [ h 1. . . . . hn)

the request is granted and the node becomes busy. With probability [1 - A ( h ( h l . . . . . hA) ] the request is thus rejected (e.g. due to a restricted transmission channel), in which case the node remains idle and restarts a new idle service, independently of the preceding one(s). Conversely, when a node h becomes busy it requires a r a n d o m amount of " b u s y service" (e.g. transmission time or packet length) with distribution function B h. U p o n completion of a " b u s y service" it requests to become idle. If upon this request other sources h I . . . . . h~

Nieo M. van Dijk received his Ph.D. in applied mathematics from the University of Leiden, The Netherlands, in 1983. Since then he has been with the University of British Columbia, the University of Twente and the Free University where he is currently associate professor of operations research. His research interests involve exact (product form) expressions, simple performance estimates (bounds) and analytic error bound (and monotonicity) results for queueing (or stochastic servicing) networks with applications in telecommunications, computer performance evaluation and manufacturing. He is co-guesteditor of a recent special issue of this journal on new telecommunications services".

N.M. van Dijk / Product forms for random access schemes

are also busy, with probability O ( h l h l . . . . . hn)

the request is granted and the node becomes idle. With probability [1 - D ( h I h a. . . . . h,)] the request is thus rejected (e.g. due to a restricted output device), in which case the node remains busy and restarts a new "busy service" independently of the preceding one(s). Further, when nodes h 1. . . . . h n are busy, then ~t'(hlh 1. . . . . h , ) ,

h ~ h l . . . . . h~,

is the service speed of idle node h, i.e., the amount of "idle service" provided per unit of time to node h, while

ever, as they naturally correspond to separate system features, we prefer not to. Corresponding queuing model

The description above can be interpreted and visualized as a two-station queuing model in which N jobs are sent back and forth, subject to state dependent blocking (reflected by A(-I" ) and D(. I" )), and with processor sharing servicing (reflected by ~/'(. [. ) and ~ ( . 1. )) at either station. D ( h l h I . . . . . hnl

~(hlh I . . . . . hol l i

• ( h i l h I . . . . . h , ) , i = 1 . . . . . n,

is the service speed of busy node hi, i.e. the amount of "busy service" provided per unit of time to node h i, i = 1 . . . . . n. Remarks

(1) As in the example below, many applications will involve only the function A(-1-) while the other functions can be set equal to 1. The inclusion of the function ~ ( - I ) may naturally arise to model a state dependent speed for transmitting, translating or processing a message at a node. The functions D(. 1- ) and xo(. I" ) do not complicate the analysis at all. They make the model totally symmetric in idle and busy nodes which can be handy for both analysis and modeling purposes. For instance, delay factors such as due to error detection (see examples 3.4 ii and 3.5 ii), service accelerations(see example 3.5 i), or message interruptions (see example 3.6) can so be modeled. (2) The assumption of a restarting idle or busy service upon blocking is common for communication systems (cf. [10,15,16,21,23,24,27,39]). Effectively it corresponds to blocked service requests to be lost. Also, as intuitively arguable in the exponential case, one can show (cf. [33,34]) that this protocol is equivalent to delaying or interrupting the idle and busy services by probability factors A ( - I ' ) or D(. I ' ) respectively. For example, a busy service (e.g. transmission of node h) will then be totally stopped whenever D ( h l . ) ---0. (3) Clearly we could have combined the functions A(-1"), g'(" l'), D(. I ' ) and ~(. I')- How-

305

[ Idlel

M

I

~

I

A(hl h t . . . . . ha')

~ ( h i l h I . . . . . hnJ] (Busy]

Examples

Below we present some examples to illustrate that the abstract model formulation given above applies to different types of communication structures. Later on (see examples 3.3) these examples will all be argued to exhibit a product form. Throughout, let H = ( h I . . . . . h , ) denote that nodes hi . . . . . h, are currently busy and write H +_ h to represent the state in which node h is added ( + ) to or deleted ( - ) from H as a busy node. Further, in these examples we assume D(-[. ) = ~ ( . I . ) = ~ ( . I . ) = 1. Example 2.1. ( C S M A ; Interference graph) (cf. [20,23,25,20]) Let the nodes in a graph represent transmitters with the restriction that adjacent nodes (neighbors) cannot transmit at the same time. In practice this is effectuated by the so-called "carrier sense multiple access" scheme (CSMA) in which a node senses the state of its channels just prior to starting a transmission and where upon sensing a busy channel as caused by a transmitting neighbor this node inhibits (aborts) the scheduled transmission. Let N(h) be the set of all neighbors of node h. Then the above description applies with A ( h l hl . . . . ' h") = { 10

ifhlotherwise.h"q~N(h)

For example, in the two-hop CSMA-figure below node 3 prohibits all other nodes to transmit at the

N.M. van D(/k / Product forms for random access schemes

306

same time, while nodes 2 and 4 can transmit simultaneously when node 3 is idle.

a structure of the form M1

5

M, 2

(~)

5

Note however, that nodes 2 and 4 which are outside hearing range can transmit simultaneously when node 3 (as well as 1 and 5) are idle. This will actually lead to a collision of their transmissions at node 3 which in turn will result in lost packets or messages. This is known as the "hidden terminal problem" (cf. [3,6,291).

Example

2.2. (BTMA) (cf [3,6,23,29,31]) To eliminate the hidden terminal problem, mentioned in example 2.1, that is the interference of nodes which are outside hearing range but which transmit to the same receiver, the so-called Busy Tone Multiple Access Scheme (BTMA) has been introduced in [29] (also see [3,6,23,31]). Under BTMA a node which hears an ongoing transmission broadcasts a busy tone signal to all its neighbors so as to prevent them from starting another transmission. For example, in the figure below, nodes 1 and 2 or nodes 1 and 7 are thus inhibited from transmitting simultaneously, as node 3 will broadcast a busy tone upon heating a message. (Hereby we assume the propagation delay for a message to travel from one node to another as negligible).

Under the BTMA-strategy, for any given set of nodes and given set of neighbors for any node, the parametrization A(h6h 1..... h , ) of example 2.1 now applies if we replace N(h) by the set of one and two-step neighbors from h. In contrast with [25], note also that a graphical representation of linked neighbors is not required.

Example 2.3. (Circuit switching) (cf [7,27]) A circuit switching network may typically have

with a predetermined fixed path P for each "source-destination" connection along which a message from that source to that destination is to be transmitted. This transmission requires one trunk from each trunkgroup along this path. Interference thus arises because of limited trunkgroups and messages using the same trunkgroups. The switches are assumed to be non-blocking (or collision free). Identify each "source-destination" connection P as a separate node which is called busy when a message along this total path takes place. Label the trunkgroups by i and let M i be the number of trunks in trunkgroup i. Then, with N~(H) the number of the ongoing messages h 1. . . . . h, in state H which use a trunk from trunkgroup i, we obtain

A( h I H) = { 10

otherwise.ifN~(H+h)~M'f o r a l l i

Example Z4. (Synchronous servicing) (cf [10,15, 16]) As a typical feature of digital transmissions, a transmission may use several time slots from a limited n u m b e r of M time slots. The following figure visualizes that a source-i message simultaneously requires bi time slots.

I

I

I

[

i

,,

bl I

I

I

I

bN

I I I

With H = (h x. . . . . h . ) representing the transmitting sources (nodes) and n i ( H ) the number among these of source type i, this is parametrized by

A(hIH) =

{loif~.bin~(H)+bh<~M i otherwise.

N.M. van Dijk / Product forms for random access schemes

(3.1)

current busy nodes is allowed to become idle again. Condition (3.2) is essential and corresponds to the property of "instantaneous attention" in the queueing literature (cf. [7,8,13,14,17]) or the condition in [4-6] that whenever a link i blocks link j then also link j blocks link i. (3) The key-condition (3.3) is related to the well-known Kolmogoroo criterion (cf. [17]) for a Markov chain to be reversible. This criterion is stated in terms of transition rates for general Markov chains. It has therefore been shown in [4-6] for multihop packet radio networks without randomized blocking (i.e. A(. I" ) having 0 or 1values only) that the Kolmogorov criterion comes down to the condition of "symmetric link blocking" (which is related to condition (3.2) above). With randomized blocking and state-dependent speeds the Kolmogorov criterion can be shown to be equivalent to condition (3.3). (Note that in (3.3) only building paths are considered.) Roughly speaking, (3.3) requires that the order in which nodes become busy is not important when regarding an additional factor A(. I ' ) ' / ' ( " I ' ) / D ( ' I ")~(" I" ) for each additional busy node.

(3.2)

Remark 3.1. (Decomposed A(. I') and dp(. I') conditions)

n A(hi, lhi ' ..... hi,_,)ff,(hiklhi, ..... hi,_,) D(hi,[hi~,.. " , hik_~)dp(hi, lh,, ' " ' ' , hi,) k1-I =l

As mentioned in remark 2.4, in various applications the functions D(. l" ) and if'(. l" ) are equal to 1. Clearly, condition (3.3) is then guaranteed if for certain functions P1(H) and P2(H):

3. Interference invariance condition In this section we will impose a concrete condition upon the system functions that will guarantee an explicit product form expression later on. To this end, let a state (h 1. . . . . hn) denote that nodes h 1. . . . . h, are busy, where h 1. . . . . h n are given in increasing order, while the other nodes are idle. The monotone ordering is introduced merely for notational convenience in the condition below but does not play any role itself. Let state JJ denote that all nodes are idle and without loss of generality assume that there exists an irreducible set H of states containing j~, i.e. a set of states such that out of any state from this set any other state within this set and no state outside this set can be reached. Condition 3.1 For any H = (h 1. . . . . hn) ~ H we have

(i)

307

for some l ~< n:

D(h, I H - ht)~(h ` I H ) > 0, (ii) for all i = 1. . . . . n:

D(h, ] H - h,)~(h, ] H ) = 0 ¢~ A( h, I H - hi)~I'( h , I H - hi) = O, (iii) There exists some value P ( H ) such that

=P(H)

(3.3)

for all permutations (i 1. . . . . in) ~ (1 . . . . . n) for which the denominator in (3.3) is positive, while P(~) is defined to be equal to 1.

fiA(hi,

lhil , .... h i , _ ~ ) = P l ( H )

(3.4)

I-I ~(h,, I h,1, .... h,, ,) = P z ( H )

(3.5)

k=l n

k=l

Discussion of conditions (1) (references [4-6]) Condition (3.1) guarantees that the product in (3.3) has a positive denominator for at least one permutation, while (3.2) guarantees that if the denominator of this product is zero then also the numerator is equal to zero, so that the product can be chosen equal to P(H). Thus effectively only permutations with non-zero denominators need to be considered. (2) Condition (3.1) could be avoided but is included as it simplifies the presentation while it excludes only the extreme case that none of the

for all permutations (i I . . . . . in) for which these products are positive. These conditions are satisfied for example, if for certain functions g(n) and h(n):

A ( h l h l ..... h n ) = g ( n ) * ( h , lh, ..... h n ) = h ( n ) .

(3.6)

Remark 3.2. (Coordinate convex interferences) An important subclass of interferences with only 0 and 1 values (i.e., no randomized blocking)

308

N.M. van Dijk / Product forms for random access schemes M1

satisfying (3.4) is obtained by •

A(hlhl ..... h . ) = { 1 0

b~

if ( h , h I . . . . . h , ) ~ C otherwise, (3.7)

M

Mi

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

(h, ..... ho)

him

C

= ( h i ..... hi-l, hi+, ..... h , ) ~ C .

(3.8)

In words that is, nodes are never blocked to become idle but the state of the busy nodes has to remain within a region C. In correspondence with [10] and [16], such interferences are called "coordinate convex". Note that the corresponding function PI(H) is equal to 1 for all H ~ C. []

When a node of type i wishes to transmit a message, it randomly hunts over b; from the M; input channels to find a free channel. With n;[H] the number of type-i messages, n the total number of messages and t h the type of node h, this is modeled by

A ( h l H ) = I ( n < M ) { 1 - ( n ; [ H ] ) ( M ; ) } b,

Examples (i = th). Below we will present several examples satisfying (3.4) and (3.5). The coordinate convex examples 3.1 have been individually studied in the literature (cf. [10,15,16]). The examples 3.2-3.4 have not been reported. Herein, all functions not specified are identical to 1.

(3.9)

The invariance condition (3.4) holds as a special example, since it holds with arbitrary functions g ( . ) and gi(') for

A(hIH )=g(n)gi(ni[H]),

and (3.10)

n;[H]

Examples 3.1 (Coordinate convex interferences) One easily verifies that the examples 2.1-2.4 are "coordinate convex" with (i) C = ( H I H has no neighbors) in example 2.1 (ii) C = ( H I H has no one or two-step neighbors} in example 2.2 (iii) C = ( H I N ; ( H ) ~< M; for all trunkgroups i) in example 2.3 (iv) C = ( H IEib;ni(H) <~M} in example 2.4.

with(i=G)

P l ( g ) = f i g ( k - 1 ) I - I I-I g ; ( z - 1). k=l

i

(3.11)

z=l

(ii) (Error detection). Consider the following circuit switching communication structure with the transmitting nodes h = 1 . . . . . M classified in four source types, let n i be the number of busy type-i sources (i.e. currently sending a message from S; to D) and t h the source type of node h.

Examples 3.2 (Randomization) In some examples the functions A(. [.) or D(. I" ) include randomization and thus have values other than 0 or 1. (i) (Random grading). The following extension of the classical "Engset ideal grading" satisfies (3.4). There are different types of nodes. A type-i node transmits type-i messages. Each message type i has an individual group of M i input channels. That is, different message types use different input channels. However, all messages share one and the same group of M output channels. A message transmission simultaneously occupies one input and output channel.

®

However, during message transmission errors may arise which are detected not before completion of the transmission. In that case the total message is to be retransmitted. For example, also referring to

N.M. van Dijk / Product forms for random access schemes

remark 3.3 below, the following error probabilities could be involved.

D(hlH) 'Pi(ni)Ql(n, + n2)Q3(nl + (t h = i ~ (1, 2)) e,(ni)Q2(n

n 2

+ n 3 + n4),

3 + n4)Q3(?'/1 q'- n 2 --[-n 3 --I--n4) ,

(t h = i ~ (3, 4)), (3.12) where P, and Qj are arbitrary functions with values between 0 and 1. The invariance condition (3.3) is then easily verified with P(H)-I

{-

/ -+P,(k) k=,[I Q,(k)

= ITI/- k=lI-I tl 3 -I~ n 4

F/I

Jr" n 2 4- ,'/3 ~- gI 4

FI Q2(k)

k=l

1-I

k=l

Q,(k). (3.13)

Remark 3.3. (Interpretation) One can note that it seems unrealistic to let the error probabilities depend on the state at the time of completion. However, one can also consider these to be the conditional probabilities of errors having occurred during transmission given that at the time of completion the system is in state H. Alternately, by interpreting the value D(. l" ) as a delay factor of obtaining service (for example thinking of transmissions to be time-slotted with D(-I" ) the error probability for one time slot which is to be retransmitted), we would obtain exactly the same expression (3.13) (In fact, in the general framework, this interpretation would come down to setting D(. 1" ) = 1 and ~/'(. 1" ) equal to the right hand side of (3.12).) This dual interpretation with exactly the same final result may seem counterintuitive. It does correspond though with a more general equivalence property as established in [34] which appears to be directly related to product form results. A similar dual interpretation will also be involved in example 3.4 (see remark 3.4).

Examples 3.3. (Delay / acceleration factors) (i) (Acceleration factors). Consider a multiprocessor with one fixed (regular) and one flexible (overflow) server for each of a finite number of input sources. A source can be serviced by both processors at double speed but overflow servers

309

are set in only when the total number of busy sources exceeds some threshold M. Then (3.5) is guaranteed by ~(hln)=

{1, 2,

n < M,

n>_.M,

(3.14)

P2 (/-/) = 2'°-MY In analogy with example 3.4 (i), the above example can be extended to type-dependent thresholds M,. More precisely, (3.5) is satisfied by substituting ~ ( - I " ) for A(- I • ) and P2(') for PI(') in example 3.2 (i).

(ii) (Delay factors) A standard delay example is a processor sharing service mechanism in which each job to be served (e.g. program to be run by a central processor unit) gets an equal share of the total capacity as prescribed by • ( h l n ) = 1/n. This can be extended to more detailed delay interferences. For example, the circuit switching example of 3.2 (ii) can be reread verbatim with D(. I" ) replaced by 4'(-1" ) representing a delay factor.

Example 3. 4. (Priority messages) In various transmission systems some messages may have "priority" over other messages in a 15reemptive manner. For example, consider a transmission device which can handle only one message at a time. Upon arrival of a "priority" message a regular transmission is interrupted and temporarily held up. Upon completion of the "priority" transmission, the regular transmission is resumed. R

P

As can be argued intuitively in the exponential case and concluded from [34], similarly to [35], in the non-exponential case (see remark 3.4 below), one can show that the steady state busy source distribution of the above system is the same under the following modified protocol. Once started, a regular transmission is continued until completion without interruptions by priority messages. The

310

N.M. van Dijk / Product forms for random access schemes

device can transmit one regular and one priority message simultaneously but, as before, a regular transmission can be started only when the device is idle while otherwise it is lost. Moreover, a regular message is to be retransmitted if upon completion of its transmission a priority message is currently transmitted.

Let R and P denote the sets of nodes that generate "regular" and "priority" messages respectively. Then under the latter modified protocol the system fits in the framework of Section 2 and satisfies the conditions (3.2) and (3.3) with n=(h,[i=l

..... M}

U { ( h i , h2) l h i ~ R and h j ~ P } ,

A(hi[H )=0

f o r h i ~ R and H 4 = ~ ,

A(hilhj)=O

forhi~Pand

D(h,[h/)=O

f o r h i ~ R and h j ~ P ,

4. Product form This section contains the main result of the paper. With reference to remark 4.4.4 below, assume that the "idle and busy service" disribution functions I h and B h have continuous density functions qh(') and f h ( ° ) with means o h and % respectively. Let the state

(S, R ) = ((sl, rl) ..... (SN, rN) ) denote that node i is idle when s~ = 1 and busy when s~ = 2 with a residual amount r i up to completion of the current idle service (s, = 1) or busy service (s, = 2) respectively, i = 1 . . . . . N. For a given node specification S = (s 1..... s,) let H be the corresponding set of busy nodes. Referring also to remark 4.4.4 below, without restriction of generality we make the following assumption.

(3.15)

Assumption 4.1. The initial conditions at time 0 and distributions I h and B h a r e such that there exists a steady state density distribution {~r((S, R))} and steady state mass distribution (~r(H)) with H restricted to H.

As a consequence, the steady state busy source product form expression (4.2) that will follow applies for the modified protocol and thus (as per the arguments above and remark 3.4 below) for the original priority protocol.

The next two theorems can then be proven. The first, of which the proof is given at the end of this section, is the technical key theorem. The second is the more practical consequence providing a simple explicit expression in which the distributional forms of the "idle and busy services" do not play a role.

hjeP

A(. I') = D(- I') = 1 otherwise, P(-)=I.

Remark 3.4. (Equivalence of interpretation in example 3.4) The claimed stationary equivalence of the original natural preemptive resume and the theoretical preemptive restart interpretetion in example 3.7 above is intuitively argued by the memoryless property in the exponential case. A formal proof in this case can be given rather directly by writing out and comparing the global balance equations. For the non-exponential case, in contrast, an equivalence of this form will not hold true generally. However, based on the product form result under the second interpretation, or more precisely, the balance principle per source as per the proof in section 5, one can conclude from [34] or [35] that in this example it does apply also for the non-exponential case. []

Theorem 4.2. Under condition 3.1 with P( H) given by (3.3), assumption 4.1 and c a normalizing con-

stant, We have for all ( S, R) with H ~ H: rr((S, R ) ) = c P ( H )

1-I [ 1 - - I h ( r h ) ] h:sh = l

×

I-I

h:Sh=2

[l--Bh(rh) 1.

Proof. See further on in this section.

(4.1) []

Theorem 4.3. Under condition 3.1 with P( H) given by (3.3) and ~ a normalizing constant, we have for

all H ~ H: ~r(H) = ? P ( H ) I-I ['rh/oh] hEH

(4.2)

N.M. van Dijk / Productformsfor randomaccessschemes Proof. First note that for arbitrary nonnegative factors f~ (.) . . . . . f s (")

~ . . . ~ f ~ ( xl) ....f.( x~ ) dx~ . . . d x N =

while by standard calculus or renewal theory it is well-known that fo°°[1 - Ih(r)] d r = o h fo~[1 - Bh(r)] d r = % . By using these observations and integrating the right hand side of (4.1) over all possible residual idle services rh for all h with sh = 1 and all busy services r, for all h with s h = 2, expression (4.2) is an immediate consequence of theorem 4.1, the fact that s h = 2 for h ~ H and s h = 1 otherwise, and the substitution c=

[]

Remarks 4.4. (1) (Expression (4.2)). Note that expression (4.2) is determined by only the mean idle and busy service requirements and P ( H ) as calculated by (3.3) in terms of concrete systems functions. (2) (Verification of condition 3.1). In principle the verification of condition 3.1 and the calculation of P ( H ) can be computationally complex. However, in most practical situations one either easily finds a counterexample with 0 and 1 values or one can recursively calculate P ( H ) as based upon "basic" paths or cycles. (Related results along this line can be found in [12], [13] and [17]. All the examples have so been verified directly. In fact, the form of A, D, • and ff~ will often lead to some conjecture for P(H). The checking of (3.3) for all paths can then be replaced by verifying

P( n ) A ( h l H ) v I ' ( h l H ) = P(H

+ h)D(hlH)t~(hlH

+ h)

for all possible states H + h, as based upon the

311

Kolm0gorov criterion for reversibility (see the discussion under condition 3.1). All the examples have so been verified directly. (3) (Normalizing constant). Another most important computational issue is that of normalization constants, in our setting that is ~ in (4.2), when the system is large. For multihop-CSMA protocols its computation is known to be NP-hard. To compute this constant, techniques such as developed in [1,16,22,25] might be exploited. Most notably, the latter reference ([25]) particularizes to interference graphs and presents an interesting approach by adopting results from statistical mechanics. As an alternative approach, to compute performance measures such as a throughput or call congestion, bounding methodologies as developed in [18] and [40] can be applied as based upon the product form result (4.2). (4) (General distributions). The assumption of absolute continuous distributions seems and in fact can be rather restrictive in practical situations. However, theorems 4.1 and 4.2 can be generalized to arbitrary distributions I h and B h such as corresponding to deterministic or partially atomic distributions, provided multiple transitions at the same time are avoided. Given that only a fixed finite number of nodes are involved, the latter condition may only, be violated by special choices of an initial distribution but will usually be satisfied. As the details of the generalization are highly technical but rather standard, below the steps will just be outlined briefly. First, one can approximate any distribution arbitrarily closely (in the sense of weak convergence) by finite mixtures of Erlang distributions (cf. [11]). Next, by considering approximate sequences of such mixtures one can also conclude weak convergence of the corresponding sample path processes on so-called D-spaces (cf. [2,11,38]). This in turn also leads to weak convergence of the associated steady state distributions (cf. [11,38]). Finally, the convergence of the corresponding expressions (4.2) to the one presented is concluded by standard analytic limiting arguments. (5) (Multiple idle and busy levels). The communication of a complex message may involve a number of busy periods with idle interruptions such as for storing or retrieving data. This can be modeled by allowing more than one type of idle and busy service per node. Similarly to [32], expression (4.2) can then be extended to an "aver-

312

N.M. van Dijk / Product forms for random access schemes

aged" version with oh and % representing the "averaged mean time" that a node is idle or busy. (6) (Arrival theorem, MVA). Similarly to [33], also the "arrival theorem" can be shown to hold provided D(-I" ) = 1. In the present setting this would read as: " T h e steady state distribution as seen by a node upon idle service completion is given "by (4.2) for the system without that node". The well-known mean value algorithm (cf. [26]) for computing performance measures, such as a throughput, as based upon the product form expression, can thus be applied. (7) (Open versions). In various standard ways (e.g. by letting N ~ oo as in (2) or by including a " d u m m y node" as in [13]), similar results can be provided to model "infinite or open" transmission systems with Poissonian inputs. (8) (Retransmission/waiting). An underlying essential assumption to the above product form result is the fact that a node can only be in two possible modes: idle or busy. Blocked transmission requests can thus not be rescheduled at a different (e.g. faster) rate or simply wait (which can be seen as an infinite rescheduling rate) until deblocking. In other words that is in our formulation blocked transmissions can be seen as lost. Roughly speaking this limitation guarantees the underlying structure to be reversible while reversibility will be violated when a node can be in more than two modes such as "idle-starved (retransmiting)-busy. However as explicit product form expressions may still remain valid also in specific non-reversible structures with blocking (cf. [12,13]) special results which do allow effects as queueing or starvation do seem possible but require more detailed investigation. []

Proof of theorem 4.2.

We need to verify the global balance or forward Kolmogorov equations. To this end, for a given state (S, R) and node i, let

(S, T) - (si, ri) + (si, Fi), denote the same state with the node i specification changed from (s,, r~) in (g/, F~). Further, we use the symbol 0 + to indicate the right hand limit at 0. Then, for a fixed state (S, R) with H representing its busy sources, the global balance equa-

tions become:

Z

0 ~r((S, R ) ) g ' ( h l H )

h:sh=1

rh)h+(2,0+)h) dP( h l H + h )D( h [ H)qh(rh)

+ rr((S, R ) - ( 1 ,

+ ~r((S, R) - (1, rh) h + (1, 0+)h) ff'(h [ H ) [ 1 - A ( h l H ) ] q h ( r h )

) +

E { ~-~hTr((S, R))qJ(hIH)

h:s,~=2

+ ~((S, R)-

(2, rh) h +(1,0r)h)

q'(hlH-h)A(h + ~((S, R)[1 - D ( h I H -

lU-h)h(rh)

(2, rh) h + (2, 0 + ) h ) ~ ( h [ H ) h)] fh(rh) I = 0.

(4.3)

It now suffices to verify (4.3) with (4.1) substituted for ~r(.). First conclude from (3.2) that for h with s h = 1 and ~/'(h I H ) = 0 or for h with s h = 2 and • ( h [ H ) = 0 all three terms within braces {... } corresponding to that node are equal to 0. From (4.1), the permutation invariant expression (3.3) for P(- ), noting that I h(0 +) = B h(0 +) = 0 and recalling that nh(') has a derivative qh('), we conclude for a node h with s h = 1: a

~rhTr(( S, R ))t1 = --qh(rh)~r((S, R) - (1, rh) h + (1, Or)h) (4.4) ~r((S, R ) -

(1, rh) h + (2, 0+)h)

A(hln)xr'(h[n) D ( h l n ) ~ ( h l H + h) (4.5) ¢r((S, R) - (1, rh) h + (1, 0+)h), provided D(h I H ) ~ ( h [ H + h) > 0. However, D ( h I H ) d P ( h l H + h ) = 0 would imply that A(h [ H ) g ' ( h I H ) = 0 by virtue of (3.2). Hence, by also assuming ~ ( h l H ) > 0 as argued above we then have A(h I H ) = 0. As a consequence, in either case and by substituting (4.4) and (4.5) the term within ( . . . } in (4.3) for h with s h = 1 is equal to ~r((S, R) - (1, rh) h + (1, 0+)h)

× qh(rh)q'(hlH){-1 +A(hlH)

+ [ 1 - A ( h l H ) ] } =0.

(4.6)

N.M. van Dijk / Productforms for random access schemes

One similarly argues that for h with s h = 2 the term within {... } in (4.3) equals ir((S, R) - (2, rh) h + (2, O+)h) xfh(rh)dP(hlH)(--1

+ D(hlH-h

+[1-D(h[H-h)]}=O,

)

313

same time. In constrast the environment of a requesting node will be taken into account in a special randomized manner. The protocol by which transmission requests are rejected will be described below.

(4.7)

regardless of whether ~ ( h l H - h )A( h I H - h) > 0 or not. We have thus verified (4.3), which completes the proof of the theorem. []

5. CSMA protocols As illustrated in Section 4, the framework of Section 2 both unifies and extends standard product form communication examples. In this section we will show that also the CSMA protocols from [23] and [3] are included and slightly generalized within this framework.

@--0--©

/1\ © Protocol For a given set of busy (i.e. transmitting) nodes H = (h 1.... h,), let B0[H ] be the number of pairs of neighbors that are both n6t transmitting and let B1[H ] the number of pairs of neighbors that fire both transmitting. Let g0(n) and gl(n) be functions such that for all states H ~ H, where H is the set of states that can arise: go(Bo[Hl)gl(Bl[Hl) > 0 and for all H + h ~ H:

5.1 Extended rude CSMA. As in examples 2.1 and 2.2 consider a set of nodes which represent transmitters, (e.g. computers, satellites). A node is said to be busy when transmitting and idle when not. While idle a transmitter h requests to start a transmission at random periods with mean o h. Its transmission or packet has a random length with mean r h. Each node h has a number of neighboring nodes N(h), i.e. nodes that can hear/receive transmissions from node h, where we assume that if node j can hear node i then also node i can hear node j. A transmission request can be rejected or granted, depending on the current neighboring environment of the transmitter. When rejected a transmission request is lost and the transmitter stays idle, that is it schedules a new request at a random time with the same mean o h. Special retransmission modes or queueing is thus not involved. The rejection will practically have involved carrier sensing of the links or carriers to determine the status of neighbors. Propagation delays are hereby assumed to be negligible, i.e. zero. When a transmission request is granted the transmission is started immediately.

Environmental inter/erence As yet, in contrast with examples 2.1 and 2.2, we do not exclude neighbors to transmit at the

g o ( B o [ H + h]) g 1 ( B I [ H + h])

a(hln)=

g0(B0[n])

gl(B,[n]) (5.1)

and that these quantities A(h I H ) are less than or equal to one. Further, for.. simplicity assume that the other functions 4 ( . l" ), q'(" l" ) and D ( - l " ) are identical to 1. The invariance condition (3.3) or rather (3.4) is then directly verified by substituting

P(H) = g°(B°[H])g'(Bl[H]) go( Bo[~j])gl(O)

,

(5.2)

noting that for all H, H + h ~ H: e ( t t + h) = e ( / 4 ) A ( h I/-/),

(5.3)

Theorem 4.3 thus applies with P ( H ) of the form (5.2).

Example: Rude-CSMA (Cf. [23]) As a special case, consider the rude-CSMA protocol from [23] given by A(hlH)

= xu~°( H)yN'~( H )

(5.4)

where N ~ ( H ) and N ~ ( H ) are the numbers of idle (not transmitting) and busy (transmitting) neighbors from h in state H and where x and y are given system parameters, with 0 < x ~< 1, 0 ~
N.M. van Dijk / Product forms for random access schemes

314

~,LOHA-protocol, x = 1, y = 0 models the stan5ard CSMA protocol of example 2.1 and other calues of x and y may reflect for instance that ;ensing of channels is not always reliable (cf. [23]). Fhe form (5.1) is now easily verified if we sub;titute

= z -B°["] l(sl[H])

=yB,W

(5.5)

'or H = H + h and H = H and noting that B o ( [ H + h]) = B o ( [ H ] ) - Nob(H)

Ba([H+ h]) = B , ( [ H ] ) + N ¢ ( H )

(5.6)

By theorem 4.2 and (5.2) the product form •esult from [23] now follows from (4.2) with

t'( H) = xB°I~IX-B°tHIy B'tH].

Further, some messages of different types by one and the same node might be transmittable simultaneously while others might not.

Example In the figure below, node 1 can transmit a message of type 1 to nodes 2, 3, 4 and of type 2 to nodes 5, 6, 7. Nodes 2, 4, 5, 7 are assumed to have t w o messages types while 3 and 6 three. N o d e 1 can hear each of these messages, so that 1 ~ Nm(h ) for all m and h = 2 . . . . . 7. Further, for illustration later on assume that node 1 can transmit only one message at a time while at nodes j 4= 1 messages of different types can be transmitted at the same time.

(5.7)

;.2 Multiple and rink-selective messages.

o

3

This example is included mainly to illustrate hat also link-selective messages as dealt with in 3] and multiple transmissions at the same time by me transmitter, as implicitly included in [3], are •overed by the general framework. As side points, t hereby directly slightly extends these results to mnexponential scheduling times, as could also rove been concluded from combining [3] with 4-6], and to selective blocking of a single mes;age-type rather than of all messages that come 'rom one transmitter. Again consider a set of nodes representing ransmitters. Now, however, we allow that a node nay transmit different messages to different sets ff neighbors. For instance, a different transmis;ion rate may be scheduled for each different mighbor or link. Say, node i can transmit a mes'.age type m to neighbors N,,(h) for m = ~,.... M(h), where the sets Nm(" ) are not required o be disjoint. The scheduling time and packet ength for different type messages of one node are ndependent, say with means oh and i-h respecively for message type m of node h.

Protocol The protocol now in order is that two neigh)ors may not simultaneously transmit a message o each other. That is, when node h transmits a nessage of type m, none of the nodes Nm(h ) can ;tart a transmission of a message to node h.

N1(1)

o

6

N2('I )

Transformation The general system described above can be transformed into the framework of Section 2 as follows. Construct a new network of transmitting nodes in which each original node is replaced by a group of nodes with a separate node for each of its message types. Exclusions of simultaneous transmissions of certain types as per the original model can now directly be presented by links. Further, the node corresponding with message type m of node h has idle and busy parameters oh and *h.

Example The transformation for the above example is illustrated below, where as above only nodes 1 . . . . . 7 are taken into account. (In fact, each end node now simply has one extra disjoint neighbor). N o d e 1 is split in two connected nodes as its messages cannot be transmitted simultaneously. N o d e 3, for example, is split in three disconnected nodes as it can transmit all message types at the same time. A type-1 transmission by node 1 prohibits any transmission at the same time by any of the original nodes 2, 3, 4 as reflected by all links in the left hand side.

N.M. van Dijk / Productforms for random access schemes

/

Remark 5.1. (Refs. [4-6]) In [4-6] the results of [3] were already generalized to more complex multihop packet radio network structures and general scheduling and transmission times. Necessary and sufficient conditions for a product form result of the above form were provided next to explicit expressions for link throughputs in different situations. Multiple transmissions by one transmitter at the same time are not included in these references.

0

0

/ N20)

N1(1)

315

Remark 5.2.

General result The original system is thus modified in the standard multihop-CSMA model from example 2.1, which satisfies the coordinate convex condition (3.7) and (3.8) (see example 3.1(i)) and thus the invariance condition (3.3) with P ( - ) = 1. Let:

Despite the simple form (5.7), note that the actual computation of a performance measure such as the throughput can still be most complicated as it requires to determine the set of admissible states H or (H, M ) and to calculate the normalizing constant. (Also see remark 4.4.3).

(H, M)= ((h, M(h)); h H}

Evaluation

be the state in which nodes h ~ H are transmitting and where node h currently transmits messages of types M(h) = {m I . . . . . rex(h) } for some x(h) and denote by (H, M ) be the corresponding state space of admissible states. Then by virtue of the above transformation of the original system into the standard multihop-CSMA system of example 2.1, and by application of theorem 4.2, the steady state distribution with ? a normalizing constant is concluded to be:

A framework is presented by which the possibility of product form results for various telecommunication packet or circuit switching random access schemes can be investigated. Exponentiality assumptions are avoided. A condition is provided, in terms of concrete system protocols, that guarantees an explicit product form expression depending upon only mean transmission times and packet lengths. This condition unifies and extends standard product form telecommunication examples, but also leads to a number of new product form examples for circuit or packet switching and resource sharing random access schemes. For instance, synchronization, random grading, error detection, delays or accelerations and priority messages can be involved. Particularly, generalizations are given of recently reported product form results for CSMA protocols. Extensions of this framework such as to include multistage or ordered transmissions seem possible.

FI

H

h~H m~M(h)

(U, M) ~ ( n , M ) .

(5.7)

Special case: single link-selective messages Assuming that a node can transmit only one message at a time, so that M(h) is always a singleton, and aggregating over the message types, we obtain:

,~(n) =~ FI ~h, H ~ H ,

(5.8)

h~H

with ~h the averaged transmission intensity for node h given by ~h

=

re(h) E "rih//aih" i=l

(5.9)

Acknowledgement The careful reading and many detailed comments of the referees, both technical and descriptive, which significantly helped to clarify the presentation are highly appreciated. Also, one of the

316

N.M. van Dijk / Product forms for random access schemes

referees is specially acknowledged for his pointing out the related results in [4-6].

References [1] 1.F. Akyildiz and N. yon Brand, Computation of performance measures for queueing networks with reversible and rejection blocking, Acta Inform., to appear. [2] A.D. Barbour, Generalized semi-Markov schemes and open queueing networks, J. Appl. Probab. 19 (1982) 469474. [3] R.R. Boorstyn, A. Kershenbaum, B. Maglaris and V. Sahin, Throughput analysis in multihop CSMA packet radio networks, IEEE Trans. Comm. 35 (1987) 267-274. [4] J. Brazio and F. Tobagi, Theoretical results on the Throughput analysis of multihop packet radio networks, Proc. ICC'84, Amsterdam, The Netherlands, July 1984. [5] J. Brazio, F. Tobagi, Conditions for product form solutions in multihop packet radio network models, SEL Technical Report 85-274, Computer Systems Laboratory, Stanford University, Stanford, CA, 1985. [6] J. Brazio, Capacity analysis of multihop packet radio networks under a general class of channel access protocols and capture models, Technical Report CSL-TR-87-318, Computer Systems Laboratory, (Stanford University, Stanford, CA, 1987). [7] D.Y. Burman, J.P. Lehoczky and Y. Lim, Insensitivity of blocking probabilities in a circuit swithing network, J. Appl. Probab. 21 (1982) 850-859. [8] K.M. Chandy and A.J. Martin, A characterization of product-form queueing networks, JACM 30 (1983) 286299. [9] R.E. Felderman, Extension to the rude-CSMA analysis, IEEE Trans. Comm. 35 (1987) 848-849. [10] G.J. Foshini and B. Gopinath, Sharing memory optimally, IEEE Trans. Comm. 31 (1983) 352-359. [11] A. Hordijk and R. Schassberger, Weak convergence of generalized semi-Markov processes, Stochastic Process. Appl. 12 (1982) 271-291. [12] A. Hordijk and N.M. van Dijk, Networks of queues with blocking, in: K.J. Kylstra, ed., Performance "81 (North Holland, 1981) 51-65. [13] A. Hordijk and N.M. van Dijk, Networks of queues. Part I: Job-local-balance and the adjoint process. Part II: General routing and service characteristics, Lecture Notes in Control and Inform. 60 (1983) 158-205. [14] A. Hordijk and N.M. van Dijk, Adjoint process, joblocal-balance and insensitivity of stochastic networks, Bull. 44-th Session Int. Stat. Inst. 50 (1983) 776-788. [15] P. Kamoun and L. Kleinrock, Analysis of finite storage in a computer network node environment under general traffic conditions, IEEE Trans. Comm. 28 (1980) 9921003. [16] J. Kaufman, Blocking in a shared resource environment, IEEE. Trans. Comm. 29 (1981) 1474-1481. [17] F.P. Kelly, Reversibility and Stochastic Networks (Wiley, New York (Chichester, UK), 1979).

[18] T. Kerola, The composite bound method for computing throughput bounds in multiclass environments, Performance Evaluation 6 (1986) 1-9. [19] L. Kleinrock and S. Lam, Packet switching in a multiaccess broadcast channel: Performance Evaluation, IEEE Trans. Comm. 23 (1975) 410-423. [20] L. Kleinrock and F.A. Tobagi, Packet switching in radio channels, Part I: CSMA models and their throughput-delay characteristics, IEEE Trans. Comm. 23 (1975) 14001416. [21] S.S. Lam, Store and forward buffer requirements in a packet switching network, 1EEE Trans. Comm. 24 (1976) 394-403. [22] D. Mitra, J. McKenna, Asyptotic Expansions for closed Markovian networks with state-dependent service rates, JACM 33 (1986) 568-592. [23] R. Nelson and L. Kleinrock, Rude-CSMA: A multihop channel access protocol, IEEE Comm. 33 (1985) 785-791. [24] R.O. Onvural, and H.G. Perros, On equivalencies of blocking mechanisms in queueing networks with blocking, Oper. Res. Lett., 5 (1986) 293-297. [25] E. Pinsky and Y. Yemini, A statistical mechanics of some interconnection networks, Performance "84 (North Holland, Amsterdam, 1984). [26] M. Reiser and S.S. Lavenberg, Mean-value analysis of closed multichain queueing networks, JACM 27 (1980) 313-322. [27] M. Schwartz, Telecommunication Networks (AddisonWesley, Reading, MA, 1987). [28] J. Swiderski, Unified analysis of local flow control mechahisms in message-switched networks, 1EEE Trans. Comm. 32 (1984) 1286-1293. [29] F.A. Tobagi and L. Kleinrock, Packet switching in radio channels: Part II: The hidden terminal problem in cartier sense multiple-access and the busy tone solution, IEEE Trans. Comm. 23 (1975) 1417-1433. [30] F.A. Tobagi and V.B. Hunt, Performance analysis of carrier sense multiple access with collision detection, Comput. Networks 4 (1980) 245-259. [31] F.A. Tobagi and D.H. Schur, Performance Evaluation of channel access schemes in multihop packet radio networks with regular structure by simulation, Comput. Networks ISDN Systems 12 (1) (1986) 39-60. [32] N.M. van Dijk, Finite source blocking systems with multi-level active and idle periods, Oper. Res. Lett. 7 (1988) 315-320. [33] N.M. van Dijk, Blocking of finite source inputs which require simultaneous servers with general think and holding times, Oper. Res. Lett. 8 (1989) 45-52. [34] N.M. van Dijk, On stop-repeat servicing for non-exponential queueing networks with blocking, J. Appl. Probab. 28 (1991) 159-173. [35] N.M. van Dijk, An equivalence of communication protocols for interconnection networks, Comput. Networks ISDN Systems 20 (1990) 277-283. [36] N.M. van Dijk and N.C. Tijms, Insensitivity in two-node blocking models with applications, in: O.J. Boxma, ed., Teletraffic Analysis and Computer Performance Evaluation (North Holland, Amsterdam, 1986) 329-340.

N.M. van Dijk / Product forms for random access schemes

[37] J. Walrand, Introduction to Queueing Networks (Prentice-Hall, Englewood Cliffs, N J, 1988). [38] W. Whitt, Continuity of generalized semi-markov processes, Math. Oper. Res. 5 (1982) 494-501. [39] Y. Yemini and L. Kleinrock, Interfering queueing

317

processes in packet broadcoast communications, IFIP Tokyo (1980). [40] J. Zahorjan, K.C. Sevcik, O.L. Eager and B. Galler, Balanced job bound analysis of queueing networks, Comm. Assoc. Comput. Machine 25 (1982) 134-141.