THROUGHPUT CALCULATIONS FOR MULTIHOP PACKET RADIO NETWORKS

THROUGHPUT CALCULATIONS FOR MULTIHOP PACKET RADIO NETWORKS

Richard A.McClain

Lockheed Martin Advanced Technology Laboratory

1Federal Street(AE-3W)

Camden,NJ08102

ABSTRACT

The main result of this paper is an ef?cient algorithm for

evaluating throughput in a multihop packet radio network

(PRN).The algorithm is needed for a Differentiated Services

Quality of Service(QoS)framework for managing the allo-

cation and policing of throughput in a PRN.But the known

algorithm for conducting analyzing throughput in a PRN is

not scalable because it requires computation on the order

of an exponential of the size of the array,and it requires

complete knowledge of the network topology.This paper

proposes and evaluates an approximation to the original al-

gorithm which scales linearly with network size and requires

only local network topology.

INTRODUCTION

The Differentiated Services architecture for network QoS is

being developed within an IETF(Internet Engineering Task

Force)working group[1].The key idea of the architecture

is to minimize the processing and storage at the core routers

which make up most of the network,by carefully control-

ling traf?c as it enters the network.Nodes at entry points

perform classi?cation,policing,and shaping to force com-

pliance to agreed?ow https://www.360docs.net/doc/f68750379.html,pliant traf?c is marked

by setting bits in the IP header.Core routers inspect these

bits when scheduling packets for forwarding.Various ser-

vices can be built up using these principles.For example,a

high priority,low delay service can be implemented by ar-

ranging for high priority traf?c to receive preferred service

compared to the rest of the traf?c which is handled as best-

effort.Making sure that the priority traf?c actually receives

low delay through the network requires a throughput model

of the network,so that the service can make rational alloca-

tion decisions,a process known as provisioning.

Based on these concepts,we are developing a QoS frame-

Figure1:QoS Framework

LINK DESCRIPTION

Consider a multihop PRN of mobile stations during an in-terval of time when the topology of the network is?xed. Stations contain a transmitter and receiver and can send and receive packets to/from other stations within range.A uni-directional communication path will be called a channel. Channels will be labeled by the combination of transmitting and receiving stations.

Stations are adjacent to those neighbor stations with which communications can occur in both directions.Bidirectional communications are a de?ned requirement for adjacency, since acknowledgements will normally be needed for reli-able communications.Also,to avoid hidden terminal inter-ference,we assume that stations employ a Request to Send (RTS)/Clear to Send(CTS)protocol and that stations which hear a CTS response will remain off the air during the inter-val when another station has been granted the clearance to send.This is the approach taken by the IEEE802.11stan-dard to avoid problems with“hidden terminals.”Let Adj be the set of stations adjacent to station.Adjacency rela-tionships are symmetric:Adj when Adj. Each(unordered)pair of adjacent stations forms a radio link. Link connects stations and bearing packets trans-mitted from or to the other.Let be the effective capacity of the link in bits per second(bps),assuming that no other station contends

for the link.The angle bracket notation will be used to indicate a bidirectional link, consisting of channels and.

Wireless networks have complex contention issues com-pared to wired networks.As a result of the link assumptions above(CSMA extended with RTS/CTS protocols)we?nd Figure2:Link’s neighbor links,consists of all links connected to stations adjacent to or including the link itself.

it easier to de?ne the rules in terms of links,since the re-quirements for sending data across a link is the same for either direction.Consider the contention rules for a link .Associated with that link is a set of neighbor links.

A link’s neighbor set,consists of all links con-nected to stations adjacent to or including the link it-self.Note that the neighbor relationship is symmetric,ie, link if link.The contention rule states that a link cannot be active simultaneously with any other link in its neighborhood.

NETWORK DESCRIPTION

Each station maintains a routing table which is used to se-lect a neighbor to forward packets to a destination.There are a number of approaches for maintaining(or generating as needed)this table,the relative merits are debated by a working group of the IETF[2].We will be assuming the ex-istence of routing tables.The complete set of routing tables for all stations is represented by the forwarding matrix, where if station forwards packets destined for station to station.Note that each row of the forwarding table corresponds to the routing table of one station.Stations only forward to adjacent stations.In order for the network to function properly,the forwarding matrix,which is dis-tributed around the network,needs to be loop free and fully connected.

AGGREGATION OF LINK FLOWS

Given a network topology,forwarding matrix,and a set of desired network?ows de?ned as traf?c sent from source sta-tions to destination stations in bits per second(bps),it is easy to compute the link?ows which would result.

Let each station request throughput for locally originated traf?c destined for other stations.Let the elements of the

network?ow request matrix,be the request for through-put from station to station.De?ne a channel distribution vector,.Element of is the quantity of traf?c on channel destined for station and can be computed as follows:

Adj

otherwise The elements of the channel?ow request matrix,can be computed by adding the elements of the channel distribution vector at each channel:

We can compute the elements of the raw link?ow request matrix,by simply adding the channel?ows associated with each direction of the link(in bps):

The normalized link?ow requests are obtained by dividing the link?ow request matrix elements by the maximum ca-pacity of the links,(also in bps):

The normalized link?ow requests can range from0(no traf-?c)to greater than1(a fully saturated link).Henceforth,the normalized link?ow requests will be referred to as just link ?ow requests.

THROUGHPUT ANALYSIS Implementing a Differentiated Services approach to QoS, requires a method for estimating available capacity of links in the PRN to carry additional traf?c.This is not an easy task,since each link contends for airtime with other links. Fortunately,an approach to modeling this complex situa-tion was developed by Boorstyn,Kershenbaum,et.al.in a series of papers from1980-87.The primary references used here are[4]and[5].They model CSMA packet radio networks as a Markovian system.The states of the Marko-vian system correspond to the combinations of active nodes in the network which can be transmitting simultaneously. The state probabilities are proportional to products of the active node sending rates and can be used to evaluate net-work throughput.But the evaluation requires calculations that are NP-hard,and although they found innovative meth-ods for speeding up the analysis,the run-times for moderate sized40node networks was measured in minutes.Also, their analysis required that all information,ie,the network topology and?ows,be gathered at a central location.

The contribution here is to implement their analytical pro-cedure ef?ciently enough for online QoS management.To adapt their analysis,a number of modi?cations are needed. First,we use the link-to-link contention rules given earlier, instead of the station-to-station contention rules used in the papers cited.Then we develop an approximation to their result,which allows the estimation of a links capacity us-ing only topology and?ows in the local vicinity of the link. This will also replace their NP-hard,exponentially complex, global,exact computation,with an approximate,scalable, distributed computation of complexity linear in the number of links.We will use their exact calculation to evaluate how well our approximation works.

The network is modeled as a continuous time Markov chain. Stations on both ends of a link attempt to initiate transmis-sions.The combined scheduling point process for link

is Poisson with rate packets/sec1.If the link is avail-able at the scheduling point,which occurs when is idle, then a packet is sent over the link whose size is exponen-tially distributed with mean length sec/packet.For convenience,we de?ne the attempted sending rate: The attempted sending rate is the fraction of the link capac-ity that would be consumed if the scheduling process always found the link available2.We must assume that all random samples are redrawn from their distributions,independent of past history.

The achieved throughput on the link,will be given by:

Prob is idle(1) The attempted sending rate satis?es whereas the achieved throughput will satisfy.

At any point in time,the network will be in a particular dis-crete Markov state,identi?ed by the combination of links which are active at that time.The states will be identi?ed by the set of active links.With links,there will be pos-sible states.Because we assume that the network protocol avoids collisions,the only states that the network reaches are the legal states which have an independent set of active links,ie,for all we have.

To evaluate the probabilities in(1),we?rst need to un-derstand the equilibrium behavior of the network by de-termining the equilibrium state probabilities,for le-gal states.Note that if the system is in state and state

is also legal,then it enters state at rate .It enters (the necessarily legal)state at rate .That observation is all that is needed to write the bal-ance equations for the steady state probability ?ow out of and into state :

These equations are satis?ed by 3:

(2)

To ?nd the probability of the null state,we use the

requirement that all state probabilities sum to 1.The sum

runs over all

which is to say for all that are independent subsets of the set of all links.(Independence was de?ned above.)

SP

Finally,from (1),we ?nd the throughput on a link is:

SP

2and

the achieved throughput on the link that depends on the at-tempted sending rates on all links (the ’s)and the topology of the network.To show that a set of throughputs is fea-sible,we must solve for the ’s which result in the desired throughputs.

To solve (5)for

’s,given a set of desired throughputs,’s,the iteration described in [5]works well.Let be the vector of attempted sending rates,.And

de?ne functions

SP SP so that (5)becomes:.We can solve for ’s by the following iteration:

If the throughputs are not feasible,then at least one of the

which is equal to the probability that the link’s neigh-bors are idle,will approach 0,so we can just terminate the

iteration if any

drops below a threshold (.001was used).Unfortunately,the evaluation of

for all re-quires the enumeration of all the legal combinations of ac-tive senders and is exponential in the number of network links.Although the papers cited above describe signi?cant progress towards optimizing the calculation,the algorithm will still not scale to networks of arbitrary size.Before we describe our approach to avoiding this problem,we ?rst de-scribe in the next section how (5)can be used to estimate the available throughput on a link.

A V AILABLE LINK THROUGHPUT

To make QoS allocation decisions,we need to estimate how much additional throughput can be sent across links in the network.As pointed out in [6],one indicator of the available link throughput is the additional fraction of time that a link could be transmitting,which we denote as .From (1),

we can express

in terms of variables from the throughput analysis:

Prob

is idle

We suggest controlling high priority allocations to maintain a minimum probability,min of each link being idle (ac-tually,just idle with respect to the transmission of priority traf?c).Now we see that all of a links time is divided be-tween being active,represented by the achieved throughput,being busy due to a neighbor link being active,denoted

as and,being idle,represented by By de?nition, The behavior of these three variables is critical to the analy-sis.What we expect is that increases in will largely be compensated with corresponding decreases in.Neigh-boring links will maintain their throughput,and the size of will be relatively inelastic to increases in a links throughput.Increasing an should cause the network to spend more time in states where more links are active simul-taneously,increasing the ef?ciency of the whole network. So generally,will decrease slightly with increasing Therefore,if we assume that does not change with in-creasing we will be taking a conservative outlook. So,agreeing to maintain a minimum idle probability,min for all links,and assuming a constant usage by neighbors, allows us to estimate the extra available throughput on a link,as:

min

ALLOCATION OF THROUGHPUT

The general plan is to assign allocations,by scaling re-quests,to keep the idle probability,above the min-imum value.(Choose,such that achieves min).The scale factor is set to unity as long as requests do not reduce the idle probability below the min-imum.The scale factor actually assigned to a link will be the minimum of all scale factors in its neighborhood.These link scale factors will then be used to develop a network scale matrix,which factors?ows depending on their source and destination.These factors determine the allocations de-livered to applications.

IMPLEMENTATION OF SUM OF PRODUCTS

CALCULATION

A major concern is the evaluation of the sum of products ex-pression(4).Our implementation of(4)is apparently sim-pler than that described in[5],although it is based on their key observation:

SP SP SP

SP

where is any member of and is the null set.This recursion allows the evaluation of each sum of products in terms of a pair of simpler sum of products.For a given num-ber of sets for which we wish to evaluate SP ie,those in(5),this recursion will expand into a much larger num-ber of SP’s,with many repeats.By creating a hash ta-ble of SP’s,indexed by their argument bitset,we create an acyclic directed graph of SP’s.In[5],they were con-cerned about choosing a proper order in which to evaluate the SP’s.We observe that since each SP depends on SP’s which have fewer bits set,we can simply order the evaluation of SP’s by the number of bits which are set to guarantee a valid order.

LOCAL APPROXIMATION

Our local approximation to the full algorithm is exceedingly simple:we compute the throughput for each link with a separate calculation of the exact same algorithm,but restricted to a subgraph of the overall link graph which in-cludes only the second neighborhood of the link.To state it precisely,to evaluate throughput on each link we con-struct a graph consisting only of links for which either (includes)or and. We have to perform the local calculation separately for each link in the network.Hence,we are replacing one large cal-culation with many smaller ones.Since we expect the com-pute time of the local calculation to be essentially constant, independent of the overall network size,we will have an al-gorithm which is linear in the size of the array and which requires only knowledge of network topology in the vicinity of a link to estimate throughput on the link.

We expect that this simpli?cation might provide reasonable estimates.Obviously,the major in?uence on the throughput of a link is its immediate neighbors which directly contend with it.The in?uence of the second neighbors on a link’s throughput are via pressure on the link’s direct neighbors to more often transmit simultaneously when they are not them-selves neighbors.The in?uence of links three edges away has much less impact.

EV ALUATION OF APPROXIMATE ALGORITHM To evaluate the approximate algorithm an experiment was conducted using randomly generated networks and traf?c models.For each test network,station locations were ran-domly placed within a square area,and nodes within a de-?ned range were designated as adjacent.The range was cal-ibrated so that stations would on average have5neighbors. Links were added if necessary to create a fully connected network.The representation of the network was an undi-rected graph with a vertex for each station and an edge for each link.Routing tables were generated using a shortest path algorithm(Floyd-Warshall[7]).The link topology was represented with a second graph,with a vertex for each link and an edge connecting each pair of adjacent links.A net-work?ow request matrix was created by randomly selecting source-destination pairs so that each station was on average originating2?ows.Both versions(original and approxi-mate)of the throughput algorithm would be run on the net-work/traf?c model and results compared.

0.2

0.4

0.6

0.8

1

-0.050

0.050.1

Absolute Error

Error Density Error Distribution

Figure 3:Absolute error in estimate of load using approxi-mate algorithm compared to the original algorithm.We conducted the comparison on 195networks (5samples for each size ranging from 10to 48stations).We measured the absolute differences in the calculated throughputs.The distribution of absolute error in throughput for all links is shown in Figure 3.The approximation tends to overestimate the original algorithm slightly,but over 95%of the time the error is less than 5%of the link bandwidth.We believe this is accurate enough for allocating throughput within our QoS framework,where some margin for error will have to be in-cluded in any case.The average runtime of these runs is shown in Figure 4.The original algorithm exhibits an ex-ponential runtime curve,and the approximate algorithm ex-hibits a linear runtime curve,as expected.Since in the QoS framework this calculation will be distributed among the QoS Managers at each node,it only requires a constant cal-culation for each node ,independent of network size,clearly a scalable algorithm.

CONCLUSION 4

A QoS framework was described and motivated the main technical body of the paper.The framework is based on providing QoS assurances to applications by aggregating re-quests and examining the overall load on the network to de-termine how much throughput can actually be allocated to individual applications.

A key enabler of the framework is the ability to estimate how much load a set of QoS throughput requests represents.For packet radio networks,we found an established prior