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NETWORK BANDWIDTH ALLOCATION PROBLEM
FOR CLOUD COMPUTING
Changpeng Yang2, Jintao You2, Xiaoming Yuan1, Pengxiang Zhao1∗
*Corresponding author
1Department of Mathematics, University of Hong Kong
2Algorithm Innovation Lab, Huawei
E-mails: yangchangpeng@huawei.com, youjintao5@huawei.com,
xmyuan@hku.hk, pengxiangzhao@connect.hku.hk
March 15, 2022
Abstract. Cloud computing enables ubiquitous, convenient, and on-demand network
access to a shared pool of computing resources. Cloud computing technologies create
tremendous commercial values in various areas, while many scientific challenges have
arisen accordingly. The process of transmitting data through networks is characterized by
some distinctive characteristics such as nonlinear, nonconvex and even noncontinuous cost
functions generated by pricing schemes, periodically updated network topology, as well
as replicable data within network nodes. Because of these characteristics, data transfer
scheduling is a very challenging problem both engineeringly and scientifically. On the
other hand, the cost for bandwidth is a major component of the operating cost for cloud
providers, and thus how to save bandwidth cost is extremely important for them to supply
service with minimized cost. We propose the Network Bandwidth Allocation (NBA)
problem for cloud computing and formulate it as an integer programming model on a high
level, with which more comprehensive and rigorous scientific studies become possible. We
also show that the NBA problem captures some of the major cloud computing scenarios
including the content delivery network (CDN), the live video delivery network (LVDN),
the real-time communication network (RTCN), and the cloud wide area network (Cloud-
WAN).
Keywords: cloud computing, bandwidth allocation, network, the 95th percentile billing,
integer programming, content delivery network, live video delivery network, real-time
communication, cloud wide area network
1
arXiv:2203.06725v1 [math.OC] 13 Mar 2022
1 Introduction
Nowadays, cloud computing is infrastructure of information technology industry;
it has boosted some revolutionary technologies in various areas and essentially
reshaped some economic ecosystems. The term “cloud” originates from the field
of telecommunications, over which providers offer Virtual Private Network (VPN)
service while redirect traffic to balance the load of the overall network is allowed,
see [1]. Cloud computing extends the VPN service; it enables ubiquitous, con-
venient and on-demand network access to a shared pool of computing resources,
which may include networks, servers, storage, and applications, see, e.g. [2].
Cloud computing is the pillar to many highly commercialized economic ecosys-
tems, and it creates tremendous commercial values.
Customers, cloud providers, and Internet service providers (ISPs) are three major
stakeholders in cloud computing. For a cloud provider, it targets to supply reli-
able, customized, and quality of service (QoS) guaranteed service for customers,
with the purpose of minimizing the bandwidth cost (paid to ISPs) which is indeed
the main component of the overall operating cost, see [3, 4]. Therefore, schedul-
ing data transmission to save bandwidth cost is extremely important for cloud
providers. In contrast to other traffic problems such as Vehicle Routing Prob-
lems, data transfer through computer networks has the following three distinctive
characteristics:
•The cost functions generated by pricing schemes are nonlinear, nonconvex,
and noncontinuous;
•Network topology updates periodically;
•Data can be replicated within network nodes.
The first feature is standard in the industry. For example, as mentioned in [5, 6, 7],
the well-known 95th percentile billing is a widely used pricing scheme, which lever-
ages 95th percentile of the bandwidth distribution over monthly periods. Due to
the variability of network status, for example, congestion may occur among some
connections, network topology may update periodically to meet requirements such
as reliability and low latency. This explains the second characteristic above. As
for the third one, digital data is a kind of special merchandise that can be repli-
cated in devices without occupying transfer bandwidth. Therefore, a network
node, which is usually referred to a server or a data center, only needs to ask
for desired data from another node while it can provide the obtained data to
more than one node. An underlying consequence is that the egress bandwidth
of intermediary network nodes contained in the path of transmitting the same
2
content is usually more than its ingress bandwidth. To the best of our knowledge,
it is the first time to consider this characteristic for modelling cloud computing
problems. With the concern of saving bandwidth cost in the industry as well as
these important attributes of data transfer over computer networks, we propose
the Network Bandwidth Allocation (NBA) problem for various cloud computing
problems, and initiate the effort of studying it from the optimization perspective.
Formally speaking, the NBA problem is defined on a network G= (V, E )dur-
ing a billing cycle P, in which V={1,2, . . . , n}is the set of network nodes
abstracted from servers or data centers; E={(i, j)|i, j ∈V, i 6=j}is the set
of directed edges abstracted from network links; and Pis a given period of
time separated by a set of sampling time points T={1,2, . . . , p}. The egress
bandwidth of the node i∈Vdenoted by φiis the summation of bandwidth
allocated to directed edges {(u, v)|u=i, v ∈V , (u, v)∈E}; and its ingress band-
width denoted by ψiis the summation of bandwidth allocated to directed edges
{(u, v)|u∈V, v =i, (u, v)∈E}. We denote by cout
iand cin
ithe admissible max-
imum egress bandwidth and maximum ingress bandwidth for the node i∈V,
respectively. During the billing cycle P, the pricing scheme is associated with the
bandwidth distribution over T. For the node i∈V, let bout
ibe the 95th percentile
of its egress bandwidth distribution {φ(t)
i}p
t=1; let bin
ibe the 95th percentile of
its ingress bandwidth distribution {ψ(t)
i}p
t=1. The cost incurred on iis given by
ui·max({bout
i, bin
i}), where ui>0is the unit-price and max(·)is the operation
to get the maximum element of a number set. Assume that in the time slot
[t, t + 1], t ∈T, the set of available edges is E(t)⊆E; the set of source nodes
providing data is S(t)⊆V; the set of destination nodes requiring data that is
originally from the source node s∈S(t)is D(t)
s⊆V\ {s}; and the bandwidth
required to transmit such data with an acceptable latency between two nodes is
at least w(t)
s>0. The NBA problem consists of determining bandwidth allocation
plans for the network Gto minimize the total bandwidth cost during P, while
the following conditions are satisfied in every time slot [t, t + 1], t ∈T:
•Data can be transmitted out from all source nodes;
•Data can be transmitted into all destination nodes;
•The egress bandwidth allocation to a node does not exceed the admissible
maximum egress bandwidth, and the ingress bandwidth allocation to it does
not exceed the admissible maximum ingress bandwidth;
•To transmit data that is originally from the same source node, the required
ingress bandwidth of a node is no more than the required egress bandwidth
unless it is a destination node.
3
Note that the last condition takes the data replication ability of network nodes
into account. The NBA problem is an abstract and generic model, and it can
be extended flexibly to suit various scenarios in practice. The NBA problem is
challenging because of its distinctive characteristics mentioned above, and also
because of the high dimensionality of its variables in real applications.
In Section 2, we review some related works. Then, we focus on the integer pro-
gramming formulation of the NBA problem in Section 3. In Section 4, we show
some real-life cloud computing applications which can be captured by the NBA
problem and its extensions. Finally, some discussions are included in Section 5.
2 Literature review
To save bandwidth cost, it is conventional to apply a centralized controller to
schedule data traffic in a network. In [8], a centralized software-driven wide area
network (SWAN) system is proposed to coordinate the sending rates and network
paths among data centers globally. Besides, a centralized traffic engineering sys-
tem named B4 is proposed in [9] to improve the utilization of Google’s global
data centers, in which traffic flows are split into multiple paths for balance. To
balance the requirements of decreasing latency and saving cost, a system called
video delivery network (VDN) is proposed in [5] to allow cloud providers to control
stream placement and bandwidth allocation dynamically. The proposed VDN in-
corporates a central controller and local agents to the traditional content delivery
network infrastructure, where the central controller generates bandwidth alloca-
tion plans based on the current network state and the local agents deal with the
incoming requests in real time. These very interesting works are essentially based
on engineering techniques, and they inspire the need of optimization models with
more rigorous theoretical analysis.
Complexity of the pricing scheme in the real world also brings challenges to sav-
ing bandwidth. The 95th percentile billing is a widely used pricing scheme of
ISPs, which is considered to reflect the required capacity of the connections in a
network better than other methods. Because most networks are over-provisioned,
there is usually reserved balance to handle bursting traffic. The 95th percentile
billing refers to ignoring the top 5% samples and charging the bandwidth used
by the leaving 95% samples. To illustrate, for a monthly billing, the samples of
consumed bandwidth are sorted at the end of each month, and the top 36 hours
(i.e., the top 5% of the overall 720 hours) of peak traffic will not be charged. It
means that the bandwidth could be used for free at a higher rate for up to 72
4
minutes per day. Therefore, mathematically the cost function generated by 95th
percentile billing is noncontinuous and nonconvex, and researchers have shown
that optimizing the burstable billing of the 95th percentile usage is NP-hard, see,
e.g. [10, 11]. To save cost charged by the 95th percentile billing, a mixed integer
linear programming (MILP) model is proposed in [6]. In [7], the authors propose
another MILP model to further relax this problem. Note that the mentioned
characteristics of transmitting data through computer networks are not fully con-
sidered in these models. For the NBA problem to be proposed, we consider the
95th percentile billing and the distinctive characteristics of data delivery through
networks. Besides, the NBA problem can be scalable to other pricing schemes if
the objective function is appropriately adjusted. Since attributes of data trans-
mission through networks are considered, feasible solutions of the NBA problem
have special properties, which will be delineated in the following sections.
3 Formulation and model
In this section, we propose the NBA problem with details, and formulate it as an
integer programming model.
3.1 Problem identification
Let the NBA problem be defined on a network G= (V, E)during a billing cycle
P, where V={1,2, . . . , n}is the set of network nodes abstracted from servers
or data centers; E={(i, j)|i, j ∈V, i 6=j}is the set of edges abstracted from
network links; and Pis a given period of time separated by a set of sampling time
points T={1,2, . . . , p}. Components are given as follows.
Definitions:
•bandwidth of an edge: the bandwidth of an edge is the amount of data trans-
mitted over it in a given amount of time, which is calculated in megabits per
second (Mbps).
•ingress bandwidth of a node: the ingress bandwidth of the node i∈Vis the
summation of bandwidth of directed edges {(u, v)|u∈V, v =i, (u, v)∈E}.
•egress bandwidth of a node: the egress bandwidth of the node i∈Vis the
summation of bandwidth of directed edges {(u, v)|u=i, v ∈V, (u, v )∈E}.
5
•ingress bandwidth capacity of a node: the ingress bandwidth capacity of a node
is its admissible maximum ingress bandwidth.
•egress bandwidth capacity of a node: the egress bandwidth capacity of a node
is its admissible maximum egress bandwidth.
•bandwidth allocation plan: a bandwidth allocation plan refers to a feasible so-
lution of the NBA problem.
•optimal bandwidth allocation plan: the optimal bandwidth allocation plan refers
to the optimal solution of the NBA problem.
Pricing scheme:
During a billing cycle P, the pricing scheme is associated with the bandwidth
distribution over T. For the node i∈V, let bout
ibe the 95th percentile of
its egress bandwidth distribution {φ(t)
i}p
t=1; let bin
ibe the 95th percentile of its
ingress bandwidth distribution {ψ(t)
i}p
t=1. The cost incurred on iis given by
ui·max({bout
i, bin
i}), where ui>0is the unit-price and max(·)is the operation
to get the maximum element of a number set.
Parameters:
•ui>0is the unit-price of bandwidth of the node i∈V.
•cin
i>0is the ingress bandwidth capacity of the node i∈V.
•cout
i>0is the egress bandwidth capacity of the node i∈V.
•E(t)⊆Eis the set of available edges in the time slot [t, t + 1], t ∈T.
•S(t)⊆Vis the set of source nodes providing data in the time slot [t, t+1], t ∈T.
•D(t)
s⊆V\ {s}is the set of destination nodes requiring data that is originally
from the source node s∈S(t)in the time slot [t, t + 1], t ∈T.
•w(t)
s>0is the least bandwidth required to deliver data that is originally from
the source node s∈S(t)between two nodes in the time slot [t, t + 1], t ∈T.
Decision variables:
f(t)
sij = 1 if an edge (i, j)∈E(t)is contained in the bandwidth allocation plan to
deliver the data that is originally from the source node s∈S(t)in the time slot
[t, t + 1], t ∈T; and 0otherwise.
6
Objective function:
The total bandwidth cost during Pshould be minimized. Denote the egress
bandwidth distribution of the node i∈Vover Tby
Bout
i={X
s∈S(t)
(w(t)
s·X
(i,j)∈E(t)
j∈V
f(t)
sij )}p
t=1,(1)
and the ingress bandwidth distribution of the node i∈Vover Tby
Bin
i={X
s∈S(t)
(w(t)
s·X
(h,i)∈E(t)
h∈V
f(t)
shi)}p
t=1.(2)
Let Q95(·)be the operation to get the 95th percentile of a set of numbers. The
goal is represented by
min X
i∈V
(ui·max({Q95(Bout
i), Q95(Bin
i)})),(3)
where max(·)is the operation to get the maximum elements of a number set.
Constraints:
•Data can be transmitted out from all source nodes in the time slot [t, t+1], t ∈T:
X
j∈V
f(t)
ssj ≥1,∀s∈S(t).(4)
•Data that is originally from a source node can be transmitted into all destination
nodes requiring it in the time slot [t, t + 1], t ∈T:
X
(i,j)∈E(t)
i∈V
f(t)
sij ≥1,∀s∈S(t),∀j∈D(t)
s.(5)
•The egress bandwidth of a node does not exceed its egress bandwidth capacity:
X
s∈S(t)
(w(t)
s·X
(i,j)∈E(t)
j∈V
f(t)
sij )≤cout
i,∀i∈V. (6)
•The ingress bandwidth of a node does not exceed its ingress bandwidth capacity:
X
s∈S(t)
(w(t)
s·X
(h,i)∈E(t)
h∈V
f(t)
shi)≤cin
i,∀i∈V. (7)
7
•To transmit data that is originally from the same source node, the required
ingress bandwidth of a node is no more than the required egress bandwidth
unless it is a destination node:
X
(i,j)∈E(t)
i∈V
f(t)
sij ≤X
(j,k)∈E(t)
k∈V
f(t)
sjk ,∀s∈S(t),∀j∈V\D(t)
s.(8)
Observation 1. As data can be replicated within network nodes, it is sufficient
to transmit a copy of data into a node by just one edge. Then, for the optimal
bandwidth allocation plan for transmitting data that is originally from the source
node s∈S(t)in the time slot [t, t + 1], t ∈T, there is
X
(i,s)∈E(t)
i∈V
f(t)
sis = 0,∀s∈S(t),(9)
and
X
(i,j)∈E(t)
i∈V
f(t)
sij ≤1,∀s∈S(t),∀j∈V\ {s}.(10)
Proof. As the node s∈S(t)already contains the transmitted data, (9) holds. To
transmit data that is originally from the source node s∈S(t)in the time slot
[t, t + 1], t ∈T, consider the node j∈V\ {s}and a bandwidth allocation plan.
If data is not transmitted over j, there is
X
(i,j)∈E(t)
i∈V
f(t)
sij = 0.(11)
If data is transmitted over j, and assume that there is
X
(i,j)∈E(t)
i∈V
f(t)
sij =m, (12)
where m≥2. Then, (12) indicates that there are medges directed to the node
j, which are denoted by (i1, j),(i2, j),...,(im, j), respectively. Select the node
ik, k ∈ {1,2, . . . , m}and delete the edge (ik, j), data can be still transmitted
into the node jthrough the remaining edges as m≥2. Besides, in the time slot
[t, t+1], t ∈T, the egress bandwidth of the node ikdecrease by w(t)
safter deletion.
Also, it holds that
Q95({φ(1)
ik, φ(2)
ik, . . . , φ(t)
ik−w(t)
s,...φ(p)
ik})≤Q95({φ(1)
ik, φ(2)
ik, . . . , φ(t)
ik,...φ(p)
ik}),
(13)
8
where φ(t)
ikis the egress bandwidth of the node ikin the time slot [t, t + 1], t ∈T.
Similarly, in the time slot [t, t + 1], t ∈T, the ingress bandwidth of the node j
decreases by w(t)
safter deletion. Also, there is
Q95({ψ(1)
ik, ψ(2)
ik, . . . , ψ(t)
ik−w(t)
s,...ψ(p)
ik})≤Q95({ψ(1)
ik, ψ(2)
ik, . . . , ψ(t)
ik,...ψ(p)
ik}),
(14)
where ψ(t)
ikis the ingress bandwidth of node jin the time slot [t, t + 1], t ∈T.
According to (3), (13) and (14), we get an equivalent or better bandwidth alloca-
tion plan after deletion. By induction, there is an equivalent or better bandwidth
allocation plan than the current one such that
X
(i,j)∈E(t)
i∈V
f(t)
sij = 1.(15)
Combining (11) and (15) yields (10).
According to Observation 1, (5) can be reformulated as
X
(i,j)∈E(t)
i∈V
f(t)
sij = 1,∀s∈S(t),∀j∈D(t)
s,(16)
which means that it is sufficient to transmit a copy of data that is originally
from the source node s∈S(t)into its destination nodes D(t)
sby just one edge
respectively.
3.2 Integer programming model
With the specifications in Section 3.1, the NBA problem can be formulated as an
integer programming model as the following:
min X
i∈V
ui·max({Q95({X
s∈S(t)
w(t)
s·X
(i,j)∈E(t)
j∈V
f(t)
sij }p
t=1), Q95({X
s∈S(t)
w(t)
s·X
(h,i)∈E(t)
h∈V
f(t)
shi}p
t=1)})
subject to X
j∈V
f(t)
ssj ≥1,∀s∈S(t),∀t∈T, (17)
X
(i,j)∈E(t)
i∈V
f(t)
sij = 1,∀s∈S(t),∀j∈D(t)
s,∀t∈T, (18)
9
X
s∈S(t)
(w(t)
s·X
(i,j)∈E(t)
j∈V
f(t)
sij )≤cout
i,∀i∈V, ∀t∈T , (19)
X
s∈S(t)
(w(t)
s·X
(h,i)∈E(t)
h∈V
f(t)
shi)≤cin
i,∀i∈V, ∀t∈T , (20)
X
(i,j)∈E(t)
i∈V
f(t)
sij −X
(j,k)∈E(t)
k∈V
f(t)
sjk ≤0,∀s∈S(t),∀j∈V\D(t)
s,∀t∈T, (21)
f(t)
sij ∈ {0,1},∀s∈S(t),∀(i, j)∈E(t),∀t∈T . (22)
To deal with this problem, the following propositions may be useful for algorithmic
design.
Proposition 1. Denote a bandwidth allocation plan for transmitting data that
is originally from the source node s∈S(t)in the time slot [t, t + 1], t ∈T, by a
graph G(t)
s= (V(t)
s, E(t)
s), where V(t)
s={i|{f(t)
sij ≥1or f(t)
sji = 1, i, j ∈V}, E(t)
s=
{(i, j)|(i, j )∈E(t), f(t)
sij = 1, i, j ∈V(t)
s}. Then, there is no cycle in G(t)
s.
Proof. We prove it by contradiction. Let G(t)
sbe the underlying undirected graph
of G(t)
s. Assume that there is a cycle contained in G(t)
s. Without loss of generality,
we denote it by
(iu, iu+1)→ · ·· → (iv, iu).(23)
Also, we denote by Vcthe set of nodes contained in the cycle (23). Then, it follows
from (10) that
X
(i,j)∈E(t)
i∈V
f(t)
sij = 1,∀j∈Vc.(24)
By (9) and (24), we have s /∈Vc, which means that nodes in Vcdo not contain
transmitted data originally. Then, to transmit data into the cycle (23) from
V(t)
a\Vc, there must be at least one node j∈Vcsuch that
X
(i,j)∈E(t)
i∈V
f(t)
sij >1,(25)
which is a contradiction to (24). Therefore, there is no cycle in G(t)
s. Moreover,
there is no cycle in G(t)
s.
10
Proposition 2. Denote a bandwidth allocation plan for transmitting data that
is originally from the source node s∈S(t)in the time slot [t, t + 1], t ∈T, by a
graph G(t)
s= (V(t)
s, E(t)
s), where V(t)
s={i|f(t)
sij ≥1or f(t)
sji = 1, i, j ∈V}, E(t)
s=
{(i, j)|(i, j )∈E(t), f(t)
sij = 1, i, j ∈V(t)
s}. Then, G(t)
sis a directed tree.
Proof. Consider G(t)
s, which is the underlying undirected graph of G(t)
s. By Propo-
sition 1, G(t)
sis an undirected graph in which any two nodes are connected by
exactly one path. Therefore, by the definition of tree, G(t)
sis an undirected tree.
Moreover, G(t)
sis a directed tree.
4 Real applications
In this section, we show some major real-world cloud computing scenarios that
are applications of the NBA problem. The scenarios include the content delivery
network (CDN), the live video delivery network (LVDN), the real-time communi-
cation network (RTCN), and the cloud wide area network (Cloud-WAN).
4.1 Content delivery network
4.1.1 Background
The content delivery network (CDN) is a geographically distributed network of
servers. The goal is to decrease latency and thus improve QoS by distributing the
service spatially relative to customers, as a shorter transmission distance usually
means lower latency [12]. Figure 1 shows the logical architecture of the CDN. For
the CDN, there are two major components:
•Server nodes of the cloud provider, whose network is a tree;
•Customers that use various devices.
For servers of the cloud provider whose network topology is a tree, there are
three categories: (1) the source server; (2) central servers; (3) edge servers. Let
the source server be the root of the tree, and there be llayers. The contents
are imported into the CDN through the source server; central servers forward
contents layer by layer; and edge servers are responsible for transmitting content
to customers. As contents can be replicated and cached within servers, it is
unnecessary to build a transmitting path from the source server to customers at
11
Source server
Layer 1
Central servers
Layer 2 to l-1
Edge servers
Layer l
Cloud provider
Customers
Figure 1: The logical architecture of the content delivery network (CDN).
all times. In other words, it only needs to build a transmitting path from a server
that already cached the desired contents to customers. Therefore, the process to
build the transmitting path for a customer can be separated into the following
two steps:
•Step 1: build a connection between this customer and an appropriate edge
server (the green arrow in Figure 1);
•Step 2: build connections from the edge server to an upper server recursively
until the desired contents are available (the red arrow in Figure 1).
Once the transmitting path is built by these two steps, the desired contents can
be delivered to the customer through this path (the blue arrow in Figure 1). For
Step 2, the path is deterministic with the given start and end, since any two nodes
of a tree are connected by exactly one path. Therefore, decisions of bandwidth
allocation are actually made at Step 1 for edge servers. The bandwidth allocation
of the source server and central servers can be estimated by the bandwidth allo-
cation to edge servers and the occurrence rate of Step 2. In the CDN, the egress
bandwidth of a server is usually greater than its ingress bandwidth. Therefore,
the amount of bandwidth for the charge is counted by the amount of egress band-
width. According to these specific attributes, we extend the NBA problem for the
CDN scenario, which is named CDN Bandwidth Allocation (CDN-BA) problem.
12
4.1.2 Formulation
Consider a CDN-BA problem which is defined on G= (V, E)during a billing cycle
P, where Gis a tree; V={1,2, . . . , n}is the set of servers; E={(i, j )|i, j ∈
V, i 6=j}is the set of edges abstracted from network links; and Pis a given period
of time separated by a set of sampling time points T={1,2, . . . , p}. Components
are given as follows.
Parameters:
•Vs⊆Vis the set of the source server and central servers; the source server is
the root node of G, and central servers are internal nodes of G.
•Ve=V\Vsis the set of edge servers; the edge servers are leaf nodes of G.
•D(t)={n+ 1, n + 2, . . . , n +m(t)}is the set of customers in the time slot
[t, t + 1], t ∈T.
•E(t)={(i, j)|i∈Ve, j ∈D(t)}is the set of available connections among edge
servers and customers in the time slot [t, t + 1], t ∈T.
•ui>0is the unit-price of bandwidth of the server i∈V.
•ci>0is the egress bandwidth capacity of the server i∈V.
•w(t)
i>0is the least bandwidth required to deliver the contents between an
edge server and the customer i∈D(t)in the time slot [t, t + 1], t ∈T.
•r(t)
i∈[0,1] is the probability that the desired content is not cached at the server
i∈Vin the time slot [t, t + 1], t ∈T.
Decision variables:
f(t)
ij = 1 if an edge (i, j)∈E(t)is contained in the bandwidth allocation plan to
transmit data in the time slot [t, t + 1], t ∈T; and 0otherwise.
Objective function:
The total bandwidth cost during Pshould be minimized. The bandwidth cost of
edge servers is measured exactly, while the bandwidth cost of central servers and
the source server in the time slot [t, t + 1], t ∈Tis estimated through r(t)
i, i ∈V.
Denote the egress bandwidth distribution of the edge server i∈Veover Tby
Bi={X
(i,j)∈E(t)
j∈D(t)
w(t)
j·f(t)
ij }p
t=1.(26)
13
Let Q95(·)be the operation to get the 95th percentile of a set of numbers. The
cost of all edge severs is represented by
X
i∈Ve
ui·Q95(Bi).(27)
Then, the bandwidth of central servers and the source server can be estimated
layer by layer. For example, let k∈Vsbe a server node whose children are edge
server nodes. Then, the egress bandwidth of kin the time slot [t, t + 1], t ∈Tis
estimated by
X
(k,i)∈E
(r(t)
i·X
(i,j)∈E(t)
j∈D(t)
w(t)
j·f(t)
ij ).(28)
For simplicity, we denote the egress bandwidth of k∈Vsin the time slot [t, t +
1], t ∈Tby b(t)
k, which is obtained by an operation Rgiven the egress bandwidth
of all edge servers and {r(t)
i}i∈V:
b(t)
k=R(k, {X
(i,j)∈E(t)
j∈D(t)
w(t)
j·f(t)
ij }i∈Ve,{r(t)
i}i∈V).(29)
Then, the total cost of central servers and the source server is denoted by
X
k∈Vs
uk·Q95({b(t)
k}p
t=1).(30)
Therefore, the goal is
min X
i∈Ve
ui·Q95(Bi) + X
k∈Vs
uk·Q95({b(t)
k}p
t=1).(31)
Constraints:
•Data can be delivered into all customer nodes from edge servers in the time slot
[t, t + 1], t ∈T:
X
(i,j)∈E(t)
i∈Ve
f(t)
ij = 1,∀j∈D(t).(32)
•The egress bandwidth of a server node does not exceed its egress bandwidth
capacity:
X
(i,j)∈E(t)
j∈D(t)
w(t)
jf(t)
ij ≤ci,∀i∈Ve;(33)
14
R(k, {X
(i,j)∈E(t)
j∈D(t)
w(t)
j·f(t)
ij }i∈Ve,{r(t)
i}i∈V)≤ci,∀k∈Vs.(34)
Remark 1. For the CDN, there are some servers that have already cached the
desired content. Once the connection between a customer and an edge server is
built, a path to transmit desired contents is established recursively. Therefore, the
constraint (4) in the NBA problem is always satisfied and thus it can be omitted
in the CDN scenario.
Remark 2. As the structure of the CDN is a tree, by Proposition 2, the constraint
(8) in the NBA problem is always satisfied and thus it can also be omitted in the
CDN scenario.
Based on the discussions above, the CDN-BA problem is formulated as
min X
i∈Ve
ui·Q95({X
(i,j)∈E(t)
j∈D(t)
w(t)
j·f(t)
ij }p
t=1)
+X
k∈Vs
uk·Q95({R(k, {X
(i,j)∈E(t)
j∈D(t)
w(t)
j·f(t)
ij }i∈Ve,{r(t)
i}i∈V)}p
t=1)
subject to X
(i,j)∈E(t)
i∈Ve
f(t)
ij = 1,∀j∈D(t),∀t∈T, (35)
X
(i,j)∈E(t)
j∈D(t)
w(t)
jf(t)
ij ≤ci,∀i∈Ve,∀t∈T, (36)
R(k, {X
(i,j)∈E(t)
j∈D(t)
w(t)
j·f(t)
ij }i∈Ve,{r(t)
i}i∈V)≤ck,∀k∈Vs,∀t∈T, (37)
f(t)
ij ∈ {0,1},∀(i, j)∈E(t),∀t∈T . (38)
4.2 Live video delivery network
4.2.1 Background
Live video means an online video that is recorded and transmitted over a network
in real-time. Delivery of live video is challenging because of the unavailability of
15
cache technology and the low latency requirement. With the increment of network
bandwidth, many applications related to live video delivery have gradually become
into reality such as live news, live shows, and live courses. Recent surveys in
[13, 14] have shown that Internet users had watched an accumulation of 1.1 billion
hours of live video in 2019 and the traffic of live video had reached 82%of overall
Internet traffic by the end of 2020. Also, it is estimated in [15] that the video
streaming market will hit 223.98 billion dollars by 2028. The diversity and volume
of live video bring not only business opportunities, but also scientific challenges
to save bandwidth cost for cloud providers.
Cloud provider Video
viewers
Video
producers
Figure 2: The logical architecture of the live video delivery network (LVDN).
Figure 2 shows the logical architecture of the live video delivery network (LVDN).
There are three major stakeholders in the LVDN: (1) video producers; (2) the
cloud provider; (3) video viewers. Video producers create live videos in various
roles such as live news reporters, actors in live shows, and teachers in live courses.
Cloud providers such as Amazon, Microsoft, Alibaba, and Huawei pay to ISPs
to deliver live video from video producers to viewers. For example, red arrows
in Figure 2 represent paths to transmit a live video from a video producer to
viewers through the cloud provider’s servers. In the process of live video delivery,
video producers and viewers demand high quality of video (e.g., video resolution,
video frame rate, color depth), instant accessibility, and low buffering ratios, while
the cloud provider targets to provide QoS guaranteed transmission service with
minimized bandwidth cost.
To cast the live video delivery problem into the NBA problem, a video producer
in the LVDN is a source node, and the related viewers are destination nodes. It
16
means that the servers of cloud providers are intermediary nodes, and the ingress
bandwidth of them is always no more than their egress bandwidth considering the
data replication ability of servers. Therefore, the amount of bandwidth for the
charge is counted by the amount of egress bandwidth. Besides, a video producer
is admissible to access only one server to upload video in this scenario. By group-
ing video producers and viewers appropriately, we extend the NBA problem for
the LVDN scenario, which is named LVDN Bandwidth Allocation (LVDN-BA)
Problem.
4.2.2 Formulation
Consider a LVDN-BA problem which is defined on G= (V, E)during a billing
cycle P,V={1,2, . . . , n}is the set of servers; E={(i, j)|i, j ∈V, i 6=j}is the
set of edges abstract from inner-domain network links; and Pis a given period of
time separated by a set of sampling time points T={1,2, . . . , p}. Components
are given as follows.
Parameters:
•S(t)={n+ 1, n + 2, . . . , n +m(t)}is the set of video producers in the time slot
[t, t + 1], t ∈T.
•D(t)
s⊆Z+\(V∪S(t))is the set of viewers of the video producer s∈S(t)in the
time slot [t, t + 1], t ∈T, where Z+is the set of positive integers.
•V(t)represents V∪S(t)∪(Ss∈S(t)D(t)
s).
•E(t)={(i, j)|i, j ∈V(t), i 6=j}is the set of available connections among servers,
video producers, and video viewers in the time slot [t, t + 1], t ∈T.
•ui>0is the unit-price of bandwidth of the server i∈V.
•ci>0is the egress bandwidth capacity of the server i∈V.
•w(t)
s>0is the least bandwidth required to deliver live video created by video
producer s∈S(t)in the time slot [t, t + 1], t ∈T.
Decision variables:
f(t)
sij = 1 if an edge (i, j)∈E(t)is contained in the bandwidth allocation plan
to deliver the live video created by the video producer s∈S(t)in the time slot
[t, t + 1], t ∈T; and 0otherwise.
17
Objective function:
The total bandwidth cost during Pshould be minimized. Denote the egress
bandwidth distribution of node i∈Vover Tby
Bi={X
s∈S(t)
(w(t)
s·X
(i,j)∈E(t)
j∈V(t)
f(t)
sij )}p
t=1.(39)
Let Q95(·)be the operation to get the 95th percentile of a set of numbers. The
goal is represented by
min X
i∈V
ui·Q95(Bi).(40)
Constraints:
•Data can be transmitted out from video producers in the time slot [t, t + 1], t ∈
T:
X
j∈V
f(t)
ssj = 1,∀s∈S(t).(41)
Note that a video producer is admissible to access only one server to upload
video in this scenario.
•Data can be transmitted into all viewer nodes in the time slot [t, t + 1], t ∈T:
X
(i,j)∈E(t)
i∈V
f(t)
sij = 1,∀s∈S(t),∀j∈D(t)
s.(42)
•The egress bandwidth of a server node does not exceed its egress bandwidth
capacity:
X
s∈S(t)
(w(t)
s·X
(i,j)∈E(t)
j∈V(t)
f(t)
sij )≤ci,∀i∈V. (43)
•The ingress bandwidth of a server node in the path to transmit data that is
originally from the same video producer is no more than its egress bandwidth:
X
(i,j)∈E(t)
i∈V(t)
f(t)
sij ≤X
(j,k)∈E(t)
k∈V(t)
f(t)
sjk ,∀s∈S(t),∀j∈V. (44)
18
Based on the discussions above, the LVDN-BA problem is formulated as
min X
i∈V
ui·Q95({X
s∈S(t)
(w(t)
s·X
(i,j)∈E(t)
j∈V(t)
f(t)
sij )}p
t=1)
subject to X
j∈V
f(t)
ssj = 1,∀s∈S(t),∀t∈T, (45)
X
(i,j)∈E(t)
i∈V
f(t)
sij = 1,∀s∈S(t),∀j∈D(t)
s,∀t∈T, (46)
X
s∈S(t)
(w(t)
s·X
(i,j)∈E(t)
j∈V(t)
f(t)
sij )≤ci,∀i∈V, ∀t∈T , (47)
X
(i,j)∈E(t)
i∈V(t)
f(t)
sij −X
(j,k)∈E(t)
k∈V(t)
f(t)
sjk ≤0,∀s∈S(t),∀j∈V , ∀t∈T, (48)
f(t)
sij ∈ {0,1},∀s∈S(t),∀(i, j)∈E(t),∀t∈T . (49)
Remark 3. The process of live video delivery can be separated into two steps: (1)
build connections between customers (video producers and viewers) and servers;
(2) build a path with the given start and end among servers. These two steps are
intrinsically related, while we may consider solving them individually in a “divide-
and-conquer” manner because it is too challenging to solve them as a whole.
4.3 Real-time communication network
4.3.1 Background
Real-time communication (RTC) is a mode of interchanging information, where
participants exchange information instantly or with negligible latency. The trans-
mitted contents include audio, videos, texts, files, and so on. Developments of
RTC technologies have boosted various application scenarios into reality, e.g.,
video chats, live conferences, smart factory, and cloud gaming. Many cloud com-
puting business have been thus created. For cloud providers supplying RTC ser-
vice, saving the bandwidth cost is also a major task, along with other purposes
such as increasing the reliability and capability of the network transmission.
Figure 3 shows the logical architecture of the real-time communication network
(RTCN). As shown in Figure 3, three customers are geographically distributed.
19
Cloud provider RTC
customers
RTC
customers
Figure 3: The logical architecture of the real-time communication network
(RTCN).
Assume that all of them are in the same RTC group (e.g., a live conference), the
cloud provider needs to build peer-to-peer paths for the customers to exchange
information. In Figure 3, example paths are labeled by arrows with different col-
ors. Besides, an RTC participant is admissible to access only one server to upload
contents in this scenario. In analogy with LVDN, RTC participants play roles
in both video producers and viewers. Therefore, the LVDN-BA problem can be
applied in an RTC group by regarding every participant as a video producer while
the remaining participants viewers. According to this idea, we extend the NBA
problem for the RTCN scenario, which is named RTCN Bandwidth Allocation
(RTCN-BA) problem.
4.3.2 Formulation
Consider an RTCN-BA problem which is defined on G= (V, E)during a billing
cycle P, where V={1,2, . . . , n}is the set of servers; E={(i, j)|i, j ∈V, i 6=j}
is the set of edges abstracted from network links; and Pis a given period of time
separated by a set of sampling time points T={1,2, . . . , p}. Components are
given as follows.
Parameters:
•S(t)={n+ 1, n + 2, . . . , n +m(t)}is the set of all RTC participants in the time
slot [t, t + 1], t ∈T.
20
•A(t)={A(t)
1, A(t)
2, . . . , A(t)
g(t)}, where A(t)
1, A(t)
2, . . . , A(t)
g(t)⊆S(t)are RTC groups
in the time slot [t, t + 1], t ∈T.
•V(t)represents V∪S(t).
•E(t)={(i, j)|i, j ∈V(t), i 6=j}is the set of available connections among RTC
participants and servers in the time slot [t, t + 1], t ∈T.
•ui>0is the unit-price of bandwidth of the server i∈V.
•ci>0is the egress bandwidth capacity of the server i∈V.
•w(t)
s>0is the least bandwidth required to deliver contents created by the
participant s∈S(t)in the time slot [t, t + 1], t ∈T.
Decision Variables:
f(t)
sij = 1 if an edge (i, j)∈E(t)is contained in the bandwidth allocation plan to
deliver contents created by the participant s∈S(t)in the time slot [t, t+ 1], t ∈T;
and 0otherwise.
Objective Function:
The total bandwidth cost during the billing cycle Pshould be minimized. Denote
the egress bandwidth distribution of node i∈Vover Tby
Bi={X
s∈S(t)
(w(t)
s·X
(i,j)∈E(t)
j∈V(t)
f(t)
sij )}p
t=1.(50)
Let Q95(·)be the operation to get the 95th percentile of a set of numbers. The
goal is represented by
min X
i∈V
ui·Q95(Bi).(51)
Constraints:
•Data can be transmitted out from all RTC participants in the time slot [t, t +
1], t ∈T:
X
j∈V
f(t)
ssj = 1,∀s∈S(t).(52)
Note that an RTC participant is admissible to access only one server to upload
contents.
21
•Data can be transmitted to other participants in an RTC group:
X
(i,j)∈E(t)
i∈V
f(t)
sij = 1,∀Ag∈A(t),∀s∈Ag,∀j∈Ag\ {s}.(53)
•The egress bandwidth of a node does not exceed its egress bandwidth capacity:
X
s∈S(t)
(w(t)
s·X
(i,j)∈E(t)
j∈V(t)
f(t)
sij )≤ci,∀i∈V. (54)
•The ingress bandwidth of a server node in the path to transmit data that is
originally from the same RTC participant is no more than its egress bandwidth:
X
(i,j)∈E(t)
i∈V(t)
f(t)
sij ≤X
(j,k)∈E(t)
k∈V(t)
f(t)
sjk ,∀s∈S(t),∀j∈V. (55)
Based on the discussions above, the RTCN-BA problem is formulated as
min X
i∈V
ui·Q95({X
s∈S(t)
(w(t)
s·X
(i,j)∈E(t)
j∈V(t)
f(t)
sij )}p
t=1)
subject to X
j∈V
f(t)
ssj = 1,∀s∈S(t),∀t∈T, (56)
X
(i,j)∈E(t)
i∈V
f(t)
sij = 1,∀Ag∈A(t),∀s∈Ag,∀j∈Ag\ {s},∀t∈T,
(57)
X
s∈S(t)
(w(t)
s·X
(i,j)∈E(t)
j∈V(t)
f(t)
sij )≤ci,∀i∈V, ∀t∈T , (58)
X
(i,j)∈E(t)
i∈V(t)
f(t)
sij −X
(j,k)∈E(t)
k∈V(t)
f(t)
sjk ≤0,∀s∈S(t),∀j∈V , ∀t∈T, (59)
f(t)
sij ∈ {0,1},∀s∈S(t),∀(i, j)∈E(t),∀t∈T . (60)
22
4.4 Cloud wide area network
4.4.1 Background
A wide area network (WAN) refers to a telecommunication network that extends
over a large geographic area. The cloud wide area network (Cloud-WAN) enables
cloud providers to supply services for clients that are geographically distributed.
The rapidly increasing demands have inspired cloud providers to focus on the
traffic exchanged between the Cloud-WAN and other networks on the Internet,
see, e.g. [7]. In cloud computing, clients access the service from the cloud through
Point of Presence (PoP). In general, the requirement of a client is responded by
the PoP which is geographically closest to the client, because a shorter distance
usually means lower latency and higher quality. However, to balance the band-
width utilization and thus save the cost, cloud providers may also redirect some
requirements to a farther PoP that can also provide service with acceptable QoS.
Cloud
PoPs
Clients
Cloud WAN
Other networks
Figure 4: The logical architecture of the cloud wide area network (Cloud-WAN).
Figure 4 shows the logical architecture of the Cloud-WAN. As shown in Figure 4,
PoPs access the cloud through inner-domain networks, while clients access PoPs
through inter-domain networks. As shown by the red arrow and the blue arrow,
the PoP that responds to the requirement of a client is different from the PoP
that received it before, which is controlled by the traffic scheduling system of the
cloud provider to achieve some central-level goals such as balancing traffic and
saving bandwidth cost. Besides, the traffic demand of a client can be transferred
by more than one PoP.
23
As the traffic transmitted into PoPs is not controlled by the cloud provider, only
the egress bandwidth of PoPs is considered in a traffic scheduling system. Accord-
ing to the NBA problem, PoPs in a Cloud-WAN are source nodes, while clients
are destination nodes. The structure of data transmission network is a bipartite
graph. By the background of Cloud-WAN, a destination node may receive data
from more than one source node. According to these specific attributes, we ex-
tend the NBA problem to this scenario, which is named Cloud-WAN Bandwidth
Allocation (Cloud-WAN-BA) problem.
4.4.2 Formulation
Consider a Cloud-WAN-BA problem which is defined on G= (V, E)during a
billing cycle P, where Gis a bipartite graph; Vd={1,2, . . . , m}is the set of
PoPs; Vr={m+ 1, m + 2, . . . , n}is the set of clients, V=Vd∪Vr;E={(i, j)|i∈
Vd, j ∈Vr}is the set of edges abstracted from network links; and Pis a given
period of time separated by a set of sampling time points T={1,2, . . . , p}.
Components are given as follows.
Parameters:
•ui>0is the unit-price of bandwidth of the PoP i∈Vd.
•d(t)
j>0is the total traffic demands of the client j∈Vrin the time slot
[t, t + 1], t ∈T.
•ci>0is the egress bandwidth capacity of the PoP i∈Vd.
•E(t)={(i, j)|i∈Vd, j ∈Vr}is the set of available connections among PoPs
and clients in the time slot [t, t + 1], t ∈T.
Decision Variables:
f(t)
ij ∈Z+represents the amount of traffic assigned to the edge (i, j)∈E(t)in the
time slot [t, t + 1], t ∈T, where Z+is the set of positive integers.
Objective function:
The total bandwidth cost during the billing cycle Pshould be minimized. As
only egress bandwidth is considered, we denote the egress bandwidth distribution
of node i∈Vdover Tby
Bi={X
(i,j)∈E(t)
j∈Vr
f(t)
ij }p
t=1.(61)
24
Let Q95(·)be the operation to get the 95th percentile of a set of numbers. The
goal is represented by
min X
i∈Vd
ui·Q95(Bi).(62)
Constraints:
•The traffic demands of a client have to be fulfilled in the time slot [t, t + 1] ,∀t∈
T:
X
(i,j)∈E(t)
i∈Vd
f(t)
ij =d(t)
j,∀j∈Vr,(63)
•The egress bandwidth of the PoP i∈Vddoes not exceed its capacity ci:
X
(i,j)∈E(t)
j∈Vr
f(t)
ij ≤ci,∀i∈Vd,(64)
Based on the discussions above, the Cloud-WAN-BA problem is formulated as
min X
i∈Vd
ui·Q95({X
(i,j)∈E(t)
j∈Vr
f(t)
ij }p
t=1)
subject to X
(i,j)∈E(t)
i∈Vd
f(t)
ij =d(t)
j,∀j∈Vr,∀t∈T, (65)
X
(i,j)∈E(t)
j∈Vr
f(t)
ij ≤ci,∀i∈Vd,∀t∈T, (66)
f(t)
ij ∈Z+,∀(i, j)∈E(t),∀t∈T . (67)
Considering the structure of (65) and (66), we quote the following proposition.
Proposition 3 (totally unimodular [16]).A matrix A is said to be totally uni-
modular if the determinant of every square submatrix formed from it has value 0,
+1, or −1. In the system of equations Ax =b, assume that Ais totally unimod-
ular and that all elements of Aand bare integers. Then, all basic solutions have
integer components.
25
5 Discussions
We propose the Network Bandwidth Allocation (NBA) problem for cloud com-
puting, and formulate it as an integer programming model. The following three
distinctive characteristics of transmitting data through networks are considered:
(1) the cost functions generated by pricing schemes are nonlinear, nonconvex
and noncontinuous; (2) network topology updates periodically; (3) data can be
replicated within network nodes. These attributes, along with the high dimen-
sionality of variables, make it very challenging to mathematically analyze the
NBA problem. The proposed NBA problem is a fundamental and high-level rep-
resentation for modeling transmitting data through networks while minimizing
the bandwidth cost, and it can be extended flexibly to suit various cloud com-
puting scenarios. Moreover, the NBA problem can be easily modified to suit for
other pricing schemes if its objective function based on the 95th percentile billing
is appropriately adjusted. We show four real cloud computing applications of
the NBA problem: the content delivery network (CDN), the live video delivery
network (LVDN), the real-time communication network (RTCN), and the cloud
wide area network (Cloud-WAN). It is expected that our first effort of nailing
down the mathematical models for these cloud computing problems can boost
more rigorous and insightful studies from various perspectives.
It is interesting yet very challenging to design efficient algorithms which can be
applied to the integer programming formulation of the NBA problem. Standard
solvers such as SCIP,CPLEX, and Gurobi turn out not to work well due to
the underlying hierarchical structures, combinatorial properties, as well as the
high dimensionality of variables of the corresponding integer programming for-
mulation. We are exploring the integration of conventional optimization tech-
niques/algorithms with problem-tailored heuristics to design applicable and/or
efficient algorithms. It is also noted that some applications usually require solv-
ing a large set of instances that are generated by homogeneous datasets yet with
different problem parameters. Hence, it is promising to apply some machine-
learning based methods to extract valuable information from data. For example,
bandwidth consumption empirically follows a regular distribution over the billing
cycles because there is a strong relationship between customer usage habits and
time over billing cycles. Besides, since the NBA problem is defined on networks,
the graph neural network (GNN) seems to be a primary deep learning architecture
that can be used to alleviate the difficulties in solving the NBA problem via its
functions of learning, reasoning, and generalizing graph-structured data.
26
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