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Note that the same data which is considered to be self-initiated by its sender, is classified as relayed by the receiving service(s). This is because that the receiving service is selected by a dynamic service composition process (reflected in P), and this selection process is not deterministic.

Markets: Uplink vs. Downlink

We consider uplink and downlink as two related markets on which data are traded. The difference between the two is that the data exchanged through the uplink market includes both self-initiated and relayed data, whereas the demands for downlink bandwidth are all relayed from the uplink market. Thus in the next section we derive one set of I-O equations for each of the two markets.

5.3.1

Network I-O Model

Given a service si ∈ S, let xi =x↑i+x↓i denote its total, uplink and downlink bandwidth costs

respectively. We now construct a model that derives these values with the limited information we have about the MSON, i.e., Θ,P,βandλ.

Definition 6. For each pair of services {si, sj} ∈S2, the elements of the uplink consumption

coefficient matrix ofS, denotedA↑= [a↑_ij]_{∣S∣×∣S∣}is given by

a↑ij=

1

βjρj

pjiβiρi (5.6)

Theorem 5.3.1. Let x↑= [x↑_i]_∣S∣×1 denote the uplink bandwidth demand vector of S, andc↑ =

[c↑_i]_∣S∣×1 denote the self-initiated demand vector of S, then when the network is in equilibrium

(meaning that each service is given the amount of bandwidth it requires to run without delay) the following equation holds

x↑

´¸¶

uplink cost

= A↑x↑

´¹¸¶

relayed uplink demand

+ c↑

´¸¶

self-initiated demand

(5.7)

is triggered by two sources, namely self-initiated and relayed. With c↑_i defined in (5.5), let h↑_ji

denote the uplink demand that is relayed fromsj tosi, i.e., whensj immediately precedessi in

an application workflow. Therefore

x↑_i= ∑

j

h↑_ji+c↑_i (5.8)

With (5.3) we derive that each run ofsj andsi is to generate data of sizeβjρj andβiρi respec-

tively. If servicesj were to be allocated an uplink bandwidth ofx↑_j, as an equilibrium entails,sj

would executex↑j/βjρj times. From the communication probability matrix P, we know that for

every one run ofsj there is a probability pji a subsequent run of si is triggered. Therefore we

have h↑_ji= x ↑ j βjρj pjiβiρi (5.8) ⇒ x↑_i= ∑ j x↑j βjρj pjiβiρi+c↑i (5.9)

Consider i ∈ {1,2,⋯,∣_S∣}, (5.9) derives the same set of equations as given by taking (5.6) into (5.7).

Definition 7. For each pair of services {si, sj} ∈S2, the elements of thedownlink consumption

coefficient matrix ofS, denotedA↓= [a↓_ij]_{∣S∣×∣S∣}is given by

a↓ij=ηji=

pjiωji

∑kpjkωjk

, sk∈S (5.10)

Theorem 5.3.2. Let x↓ = [x↓_i]_∣S∣×1 denote the downlink bandwidth demand vector of S, then

when the network is in equilibrium (meaning that each service is given the amount of bandwidth it requires to run without delay) the following equation holds

x↓

´¸¶

downlink cost

= A↓x↑

´¹¸¶

relayed downlink demand

(5.11)

Proof. It is easy to understand that within the MSON, the downlink cost is totally dependent on the uplink cost in the sense that no receive action is required if no data was sent, and that all data sent by a service in context of the MSON must be received by another service of the MSON. On this basis, leth↓_ji denote the downlink cost relayed from data sent fromsj tosi, i.e.,

5.3 The Economy of Mobile Service-Oriented Networks

the amount of data sent fromsj tosi, and we have

x↓_i= ∑

j

h↓_ji (5.12)

Recall from (5.4) that the probability that a unit of data sent by si to sj is given by ηij, we

derive h↓_ji=x↑_jηji (5.12) ⇒ x↓_i= ∑ j x↑_j pjiωji ∑kpjkωjk , sk∈S (5.13)

Similarly to the proof of theorem 5.3.1, by enumerating (5.13) with i∈ {1,2,⋯,∣S∣}, we get the same set of equations as given by taking (5.10) into (5.11).

Observe that from the definition given by (5.2), we know that ωji=0 wheni=j, therefore

the entries on the main diagonal ofA↓ are all zero. In contrast, the main diagonal ofA↑ are not necessarily all zero. These properties of the two coefficient matrices match the behaviour of a ser- vice in a practical sense. When a service (recursively) calls on itself (refer to the communication pattern given by [103] and [104]), the pair of send and receive action itself is local and thus does not cost the hosting device’s bandwidth to the access network. However, the consequence of this communication does not exclude the possibility of data being produced by the newly invoked service call. This new data has a non-zero possibility (if ρi >0) to be destined to services that

are not locally available, and thus would incur a cost to the hosing device’s uplink bandwidth. To conclude the network I-O model, we gather the per-service cost from both markets and derive the total bandwidth cost for a host devicem∈_Mas

bm=b↑m+b↓m= ∑ i x↑_i+ ∑ i x↓_i = ∑ i xi, Θ(si) =m (5.14)

withbm,b↑m andb↓mdenote the total, uplink and downlink bandwidth cost of m.