ENHANCING AVAILABILITY FOR DISTRIBUTED REPLICATED SERVICES CONSIDERING NETWORK EDGE AVAILABILITY

International Journal of Computer Networks & Communications (IJCNC) Vol.11, No.1, January 2019
DOI: 10.5121/ijcnc.2019.11101 1
ENHANCING AVAILABILITY FOR DISTRIBUTED
REPLICATED SERVICES CONSIDERING NETWORK
EDGE AVAILABILITY
Manghui Tu1
, Liangliang Xiao2
and Dianxiang Xu3
1
Department of CITG, Purdue University Northwest, Hammond, Indiana
2
Department of CSIT, Frostburg State University, Frostburg, Maryland
3
Department of Computer Science, Boise State University, Boise, Idaho
ABSTRACT
Mechanism to improve data or service availability is critical for an enterprise to ensure the quality of
service in terms of availability. Replication has been used to improve system availability. The number and
location of the replicas are two impact factors on availability. In this paper, we will consider the impact of
the node and network edge failures on the availability of replicated data or services. The Effective
availability modeling approach is designed and efficient availability computing algorithms are developed
to model and compute availability of replicated services for systems with the tree topology. The availability
enhancement problem (maximizing the objective function) is transformed to a p-median problem
(minimizing the objective function) through re-define the availability enhancement problem. An efficient
replica allocation algorithm is developed to improve data availability in tree networks, with a runtime
complexity of O(K|V|2
), where K is the number of replicas and |V| is the number of nodes in the tree
network. Finally, experimental studies have been conducted to evaluate how efficient and effective the
proposed availability computing algorithm and the availability enhancement algorithm on improving the
availability of replicated data or services. The results show that the proposed solutions are efficient and
effective on availability computing and availability enhancement.
KEYWORDS
Network Protocols, Wireless Network, Mobile Network, Virus, Worms &Trojon
1. INTRODUCTION
It is well known that the utility of the data and service is mainly limited by availability [1, 2, 3, 4],
and the dynamic environment and unreliable internet connectivity raise the concerns on
availability of distributed systems [5, 6, 7, 8, 9]. Replication and erasure coding techniques [7, 8,
9, 10, 11, 12, 30] have been used to enhance data availability. However, the data object that is
replicated also impacts the effective availability. Replicating data at highly available sites will
result in high data availability. Also, widely distributing data can result in higher availability than
distributing data in local clusters. In [8, 9], it has shown that even 100% local availability does
not necessarily provide high availability to end users. The network failures may, for 1.5-2 %of the
time, prevent clients from successfully accessing a cluster [5, 6, 7, 13]. Thus, the location of data
stored in the system plays a significant role in overall system availability.
Most of the existing research on system availability consider the availability of the storage nodes
only [11, 12, 14]. Some other research also considers the impact of data consistency on service
availability [8, 9]. All these research do not consider the reach availability of users’ requests to
service/data sites. As discussed earlier, availability of the network links also impacts service data

2
availability, especially the reach availability of users’ requests. In [6, 7], network link availability
is considered, but the evaluation algorithm itself requires exponential computation time relative to
the number of storage nodes even in a tree graph with a single replica. Thus, it is not feasible to
compute the availability of distributed systems with replications. Also, the work in [6, 7] does not
provide an availability model in a system with more than one replica, i.e, the system has multiple
service/data sites available to serve a user’s request.
In this paper, we consider modeling the service/data reach availability (i.e., the availability of
service/data to remote requests) in distributed systems in a tree network. Data or services are
replicated in the system to improve both performance and availability. Since the system has
multiple service/data sites available to serve a user’s request, a single site failure does not lead to
the unavailability to user’s request. Therefore, our problem is fundamentally different from
network reliability problem (reliable only ensures the access to a single site) [15]. To assess
system availability, efficient algorithms are developed to compute system availability considering
both node and link failures.
The remainder of the paper is organized as follows. System modeling is described in Section II.
An algorithm on computing system availability in a tree network is presented in Section III.
Section IV, issues on how to reduce the runtime complexity and how to improve the service/data
availability will be discussed. Efficient replica allocation algorithms to improve system
availability will be introduced in Section V. Experimental studies conducted will be described in
Section VI to give some numerical results of the proposed replica allocation algorithm applied in
a system with tree network topology. Section VII briefly discusses some related works in
availability modeling and enhancement and Section VIII concludes the paper.
2. SYSTEM & AVAILABILITY MODELING
The Study shows that user access requests usually follow a constant path routing to data sites and
most routes are stable in days or even weeks [16]. So the data access routes in such a distributed
system can be modeled as a tree graph, T = (V, E), as shown in Fig.1. Let e(u, v) denote the edge
that directly links two nodes u and v, and (u, v) denote a single path between u and v. Note that
e(u, v) itself is a path between u and v with a single edge. The system hosts a set of data objects
replicated to different nodes. For simplicity, we consider a single data object d. The set of nodes
hosting replicas is defined as the residence set, R = {u| u  V and u holds a replica}, and |R| is the
number of replicas. A read request needs to access the closest node to the request. If it is not
available, then, the request will be further forwarded to another node. Let r denote a read request
and Ar
(u) denote the number of read access requests from a node u over a time period. Majority
of the user accesses are reading accesses, update accesses are not considered in this research.
u
v
’
Fig.1 A sample tree network with replica.
In a distributed system, data or service availability may be affected by other factors such as node
up/downtime, system load, data consistency, and network failure or congestion [5, 8, 9].
According to [8, 9], weaker consistency would result in higher availability. Here, we assume that
users are allowed to read stale data and, hence, the effect of data inconsistency will have no
impact on data availability. Also, due to replication, the overloading problem is not an issue,
hence, the impact of system load on data availability is not considered.

3
2.1. Data Availability Analysis
If the preferred node in R is not available, then the read request will be forwarded and served at
the next available node. This process repeats until it is finally served by one of the replicas, or
finally none of the replica can serve the read request, by choosing different combinations of paths.
The access path of such a read request can be modeled as a directed graph, and thus both the
network topology and the location of the read requests originally issued will affect data
availability. Therefore, to compute data availability, we need to consider requests originated from
different nodes.
Let SA(u, G, R) denote the availability of d for a request r issued at node u with resident set R in
G. If u hosts a replica, then d is available locally with probability NA(u). When data d is
unavailable on u, a read request r needs to be forwarded to other nodes in R. Suppose that r is
forwarded to v, where v  R. The availability of data d on v to r can be computed as the product
of NA(v) and the availability of path (u, v), denoted as A(u, v). The availability of path (u, v) is
computed as the product of the availabilities of all the edges in (u, v), i.e., A(u, v) =
( , ) ( , )
( , )ee x y u v
A x y . As long as there is a node y such that y  R and the path (u, y) is
available, the request can be served. If the availabilities of different paths are independent from
each other, data d is unavailable for a request r issued in u only if all those nodes hosting replicas
are unavailable. Therefore, the availability of a request issued from node u in G can be computed
as the product of the node availability and path availability from node u to each node in resident
set R, e.g.,
SA(u, G, R) = 1 – (1 ( , ) ( ))Ay R
A u y N y
  (1)
2.2. Data Availability Modeling in a Tree Network
Consider a request originated from node u in the system with a tree graph T = (V, E). Based on
the user access protocols described, the tree network should be rooted at u, which is shown in Fig.
2. Consider a subtree Tw rooted at w, which is a subtree of T rooted at u (as shown in Fig.2). Data
object d is not available for a request r issued by u if and only if d is not available at w or any of
its child nodes. Let SUA(u, Tw, R) and SA(u, Tw, R) denote the unavailability and availability of data
object d in Tw for request r. Let ch(w) denote the set of children of w. Then,
SUA(u, Tw, R)=(1–NA(w)) * ( )
(1 ( , ) ( ))Aei ch w
A w i N i
  .
u
w
w1 w2 wi
Tw Tw’ Tj
v
w’
Fig.2. A sample tree graph rooted at u.
The unavailability of Tw’, …, and Tj, can be computed in the same way as Tw. The unavailability
of these subtrees can be applied to the computation of the unavailability of the subtrees rooted at
their parents. For example, the subtree rooted at v can be computed as SUA(u, Tv, R) = (1 – NA(v))
* (1 ( , ))*(1 ( , ) ))e UA w
A v w S u T R  * '
(1 ( , '))*(1 ( , , ) ))e UA w
A v w S u T R  . For a request r issued in u, the
unavailability of data d in the system can be computed recursively and, hence, SUA(u, Tu,
R) = (1– NA(u)) * ( ) )(1 ( ( , )*(1 ( , , )))e vUAv child u A u v S u T R   .

4
Subsequently, SA(u, Tu, R) = 1 – SUA(u, Tu, R). The data availability for a request from another
node x, e.g., SA(x, Tx, R) (note that the tree should now be rooted at x) can be computed in a
similar way. Based on the analysis above, the system availability of d is computed as the average
availability for requests in the entire system. Let SA(G, R) denote the system availability of data d.
Then, we have,
SA(G, R) = ( ( ) ( , , )) / ( )
r r
A xx V x V
A x S x T R A x 
  (2)
The algorithm Tree_SA(T, R), which is developed to compute SA(T, R) in a tree graph T = (V, E) is
shown in Fig. 3. The algorithm works as what follows. It calls the recursive algorithm T_SUA(u,
Tu, R) for each node u in T to compute the unavailability for request issued at node u. The
algorithm T_SUA(u, Tu, R) recursively compute the data unavailability for the tree rooted at node u
in T, by traversing each link and each vertex once. The computation uses the models we proposed
earlier. Note that the children of node v are the nodes in the new tree rooted at node u (There are
different implemetnations on building a new Tu in the original tree T, e.g., 1) physically re-build
the tree rooted at Tu; 2) build a new virtual tree Tu by tracking the neighboring nodes for each
node v in T, such that the unvisited neighboring nodes are child of v and the visited neighboring
node is the parent node of v in new tree Tu.
Tree_ SA(T, R) {
totalAvai = 0; SUA(u, R) = 0; totalRead = 0;
For  u  V{
SUA(u, Tu, R) =T_SUA(u, Tu, R);
SA(u, Tu, R) = 1– SUA(u, Tu, R);
totalAvai+= Ar
(ux) * SA(u, Tu, R);
totalRead += Ar
(u);}
SA(T, R) =totalAvailable / totalRead;}
T_SUA(u, Tv, R) {
ch(v) = {child nodes of v }; //all neighboring nodes
if (ch(v) == ) SUA(v, Tv, R) = 1– NA(v);
else SUA(u, Tv, R) = (1 – NA(v)) *
( )1 ( ( , ) (1 _ ( , , )))e UA ii ch v A v i T S i T R    ;
return SUA(u, Tv, R); }
Fig. 3 . The algorithm Tree_SA(T, R) in tree graph T .
In Theorem 1, we show that the availability d in tree T, SA(T, R), is monotonically increasing.
Theorem 1. Let R1 and R2 denote two resident sets of T and R1  R2, then SA(T, R1)  SA(T, R2).
Proof: Consider R1 = R2. According to the definition of SA(T, R1) and SA(T, R2), we know that
SA(T, R1) = SA(T, R2).
Consider R1  R2. For each request in a node u, it has at least one more node that can provide a
replica of data object d to access. Without loss of generality, assume that |R2| =|R1| +1, and R2 = R1
 {v}. According to the availability model, the unavailability of v, 1− NA(v) < 1. According to the
availability computing model, for each request r issued at node u, which means SUA(u, Tv, R2) <
SUA(u, Tv, R1), or SA(u, Tv, R2) > SA(u, Tv, R1). Thus, we have SA(T, R1) < SA(T, R2). It follows that
SA(T, R1)  SA(T, R2) if R1  R2.
We now show that the runtime complexity of our proposed availability computing algorithm in a
tree T is in the order of the square of the number of nodes in T.

5
Theorem 2. The runtime complexity of algorithm Tree_SA(T, R) is O(|V|2
).
Proof. It suffices to prove the correctness of recursive relations.
Let vj be a leaf node. If vj{rj
1
, …, rj
i
} or q  0, the value of G(vj,q,rj
i
) should not be counted
since the situation is not included in the definition. Therefore we set G(vj,q,rj
i
)=∞. On the
contrary, if vj=rj
s
{rj
1
, …, rj
i
} and q  1, G(vj,q,rj
i
) equals to the optimal value of the sub-problem
defined on the sub-tree Tj, which is the node vj. Therefore G(vj,q,rj
i
) = Ar
(vj)Aj
s
.
If rj
i
Vj or q < 0, the value of F(vj,q,rj
i
) should not be counted since the situation is not included
in the definition. Therefore we set F(vj,q,rj
i
)=∞. If rj
i
Vj and q = 0, the leaf node vj has to fetch
data from node rj
i
. Therefore F(vj,q,rj
i
)=Ar
(vj)Aj
i
. If rj
i
Vj and q  1, the leaf node vj can either
fetch data from node rj
i
or fetch data from itself. Therefore F(vj,q,rj
i
)=max{F(vj,0,rj
i
),G(vj,1,vj)}.
Let vj be an internal node. If Vj∩{rj
1
, …, rj
i
}= or q  0 or i = 0, the value of G(vj,q,rj
i
) should not
be counted since the situation is not included in the definition. Therefore we set G(vj,q,rj
i
)=∞. If
rj
i
Vj, at least one node must be selected in {rj
1
, …, rj
i
} ∩ Vj is equivalent to at least one node
must be selected in {rj
1
, …, rj
i1
} ∩ Vj. Therefore G(vj,q,rj
i
)=G(vj,q,rj
i1
). The rest situations can be
separated into 3 cases: rj
i
=vj and q  1, rj
i
Vj1 and q  1, and rj
i
Vj2 and q  1. We discuss them as
follows.
In case 1 we consider that rj
i
=vj and q  1. Without loss of generality, we assume that rj
i
corresponds to rj1
i1
for vj1 (the left child node of vj) and rj2
i2
for vj2 (the right child node of vj).We
claim that
G(vj,q,rj
i
)=max{G(vj,q,rj
i1
),Ar
(vj)Aj
i
+max{F(vj1,q1,rj1
i1
)+F(vj2,q2,rj2
i2
)| 0q1,q2q1, q1+q2=q1}}.
In order to prove that, we must show (1) the optimal solution appears either in G(vj,q,rj
i1
) or
Ar
(vj)Aj
i
+max{F(vj1,q1,rj1
i1
)+F(vj2,q2,rj2
i2
) | 0q1,q2q1, q1+q2=q1}, and (2) the optimal solution
is the maximum value.
First, if a node in {rj
1
, …, rj
i1
} ∩ Vj is selected in the optimal solution, then the optimal solution
appears in G(vj,q,rj
i1
). Otherwise we assume that no node in {rj
1
, …, rj
i1
} ∩ Vj is selected but rj
i
is selected in the optimal solution. Then it is better for vj to fetch data from node rj
i
=vj than other
nodes. Let q1 nodes will be selected from Tj1 and q2 nodes will be selected from Tj2 where
q1+q2=q1. Hence the optimal solution appears in Ar
(vj)Aj
i
+max{F(vj1,q1,rj1
i1
)+F(vj2,q2,rj2
i2
) |
0q1,q2q1, q1+q2=q1}.
Second, Ar
(vj)Aj
i
+max{F(vj1,q1,rj1
i1
)+F(vj2,q2,rj2
i2
) | 0q1,q2q1, q1+q2=q1} may contain some
inconsistent values. For example, if F(vj1,q1,rj1
i1
) or F(vj2,q2,rj2
i2
) selects a node in {rj
1
, …, rj
i1
} ∩
Vj, vj should fetch data from that node instead of rj
i
. However since the term Ar
(vj)Aj
i
is used, it
actually counts that vj fetches data from node rj
i
. Or F(vj1,q1,rj1
i1
) selects a node such that vj2
should fetch data from that node outside Tj2 instead of rj2
i2
. However the term F(vj2,q2,rj2
i2
)
indicates that actually it counts that vj2 fetches data from node rj2
i2
outside Tj2. Fortunately, the
inconsistent values are smaller than the corresponding consistent values, and hence are smaller
than the optimal solution. Therefore the inconsistent values will not affect the optimal solution to
be the maximum value.
In case 2 we consider rj
i
Vj1 and q  1. Without loss of generality, we assume that rj
i
corresponds
to rj1
i1
i2
for vj2 (the right child node of vj). We claim that
G(vj,q,rj
i
)=max{G(vj,q,rj
i1
),Ar
(vj)Aj
i
+max{G(vj1,q1,rj1
i1
)+F(vj2,q2,rj2
i2
) | 1q1q, 0q2q, q1+q2=q,
the associated set of G(vj1,q1,rj1
i1
) does not contain rj
1
, …, rj
i1
but contains rj
i
}}.
1
, …, rj
i1
i1
1
, …, rj
i1
i

6
is selected in the optimal solution. We claim that vj will not be selected. Because if vj is selected,
then it cannot be in {rj
1
, …, rj
i1
} ∩ Vj. Thus it is better for vj to fetch data from node rj
i
than vj.
Consequently it is better for all nodes to fetch data from node rj
i
than vj. It implies that the
selection of vj will not gain any benefit. Therefore vj will not be selected. So q1 nodes will be
selected from Tj1 and q2 nodes will be selected from Tj2 where q1+q2=q. Hence the optimal
solution appears in Ar
(vj)Aj
i
+max{G(vj1,q1,rj1
i1
)+F(vj2,q2,rj2
i2
) | 1q1q, 0q2q, q1+q2=q, the
associated set of G(vj1,q1,rj1
i1
1
, …, rj
i1
but contains rj
i
}.
Second, if the associated set of G(vj1,q1,rj1
i1
1
, …, rj
i
, then that G(vj1,q1,rj1
i1
)
should not be counted. If the associated set of G(vj1,q1,rj1
i1
) contains rj
1
, …, rj
i1
, then the
inconsistent value is smaller than the corresponding consistent value, and hence smaller than the
optimal value. so that G(vj1,q1,rj1
i1
) can be omitted. Therefore it suffices to consider G(vj1,q1,rj1
i1
)
whose associated set does not contain rj
1
, …, rj
i1
but contains rj
i
.
i
i
corresponds
to rj1
i1
i2
for vj2 (the right child node of vj). Analogous to the
proof of case 2, it can be proved that
G(vj,q,rj
i
)=max{G(vj,q,rj
i1
),Ar
(vj)Aj
i
+max{F(vj1,q1,rj1
i1
)+G(vj2,q2,rj2
i2
) | 0q1q, 1q2q, q1+q2=q,
i2
1
, …, rj
i1
but contains rj
i
}}.
If rj
i
i
i
)=∞. If rj
i
Vj and q  0, without loss of generality,
we assume that rj
i
corresponds to rj1
i1
i2
for vj2 (the right
child node of vj). We claim that
F(vj,q,rj
i
)=max{G(vj,q,rj
i
),Ar
(vj)Aj
i
+max{F(vj1,q1,rj1
i1
)+F(vj2,q2,rj2
i2
) | 0q1q, 0q2q, q1+q2=q}}
First, if the optimal solution selects node in {rj
1
, …, rj
i
} ∩ Vj, then the optimal solution appears in
G(vj,q,rj
i
).
Otherwise we assume that no nodes in {rj
1
, …, rj
i
} ∩ Vj is selected in the optimal solution. Then it
is better for vj to fetch data from node rj
i
than other nodes. Additionally vj will not be selected
because of the same reason in the proof of case 2. So q1 nodes will be selected from Tj1 and q2
nodes will be selected from Tj2 where q1+q2=q. Hence the optimal solution appears in
Ar
(vj)Aj
i
+ max{F(vj1,q1,rj1
i1
)+F(vj2,q2,rj2
i2
) | 0q1q, 0q2q, q1+q2=q}.
Finally, the inconsistent values in Ar
(vj)Aj
i
+max{F(vj1,q1,rj1
i1
)+F(vj2,q2,rj2
i2
) | 0q1q, 0q2q,
q1+q2=q} are smaller than the corresponding consistent values, and hence are smaller than the
optimal solution. Therefore the inconsistent values will not affect the optimal solution to be the
maximum value.
3. SYSTEM AVAILABILITY ENHANCEMENT
In this research, we proposed to develop mechanisms to optimize data availability in a system
with a fixed number of replicas, K, i.e., to find a resident set with a size of K such that the overall
data availability in a tree graph is optimal. It has been shown in our previous work in [17, 18] that
reducing average distance of the paths (number of hops) from user’s requests to service/data sites
can result in high availability, as the shorter the distance (the smaller the number of hops), the
higher the path availability from a requester to its closest replica. Therefore, a potential approach
to improve the overall system availability is to allocate data replicas in the system such the
average distance of the paths from user’s requests to service sites is minimal.

7
3.1 Availability Enhancement Problem Modeling
The replica allocation problem to optimize data availability is much more complex than the
traditional optimal resident set problems [19, 20, 21]. For traditional resident set problem, each
request will be served by a single site (usually the closest site) and the corresponding
communication cost will be computed as the final communication cost for such a request.
Therefore, each request will be associated with a single data replica site only. However, with the
availability model defined in this research, a request may need to find another available site if the
current accessing site is not available. This process will repeat till such access is served by a
replica site or is not served after all replica sites have been tried but none of them are available.
Thus, each request should be associated with all sites within the resident set for data availability
computing. To find the resident set with maximal data availability in a tree network, the
computation complexity could be too high to be feasible. For example, an intuitive approach is to
try every combination of the resident set in the tree and the runtime complexity would be |V||R|
.
Thus, an approximate solution is to apply the property observed in [17, 18] that the smaller the
number of hops from a request to the resident set, the higher the data availability.
First, since a request will be forwarded to a secondary replica only the primary replica node
cannot serve such request, and such an event has a very low probability (node availability is
usually very high), we can assume that a request is only associated with the primary replica node
to reduce the complexity (note that this assumption is only valid for replica allocation purpose, a
request is still associated with all replicas when computing availability). Thus, our problem can
be reduced to a resident set problem maximizing the total of the product of the number of reading
requests and the path availability of each request to the resident set. i.e.,
Σνεv(Aγ
(ν)*(Aδ(ν,R)*NA(ν))), where ( , )A v R
is the path availability from a request issued at
node v to a replica site in the resident set R.
Second, we all understand that the heterogeneity of the node (replica site) availability can have a
big impact on the overall system availability. Here, we will show that the heterogeneity of the
edge availability can have big impact on the overall system availability. In a system with line
topology shown Fig. 4, the allocation of two replicas to optimize the availability without
considering the heterogeneity of edge availability is shown in allocation scheme A, and the
allocation considering the heterogeneity of edge availability is shown in allocation scheme B
(with homogeneous node availability of 0.999). In Scheme B, the service to requests at node v1 is
available locally with availability of 0.999, and requests need to be forwarded to node v3 only if
node v1 is not available (with a probability of 0.001). Requests at other nodes can be served by the
replica at node v3 with very high probability, and only need to be forwarded to node v1 if node v3
is not available or edge e(v2, v3) is not available (for requests from node v2). The overall
normalized data availability in system with replica allocation scheme B is around 0.96 by
applying the availability computing algorithm described in Fig. 3. In Scheme A, the service to
requests at node v1 is not available locally and thus need to be forwarded to node v2 or v3 where
the availability of edge e(v1, v2) is 0.5. The overall normalized data availability in system with
replica allocation scheme A will definitely be lower than 0.90.
v1
0.5
v5v3
0.9 0.9 0.9
v2 v4
0.5
v5
0.9 0.9 0.9
Allocation Scheme A (each node issues 10 requests)
Allocation Scheme B (each node issues 10 requests)
v2v1 v3 v4
Fig. 4. The impact of the edge availability on system availability

8
Based on the above analysis, the availability optimization problem considering the heterogeneity
of network link availability is desired to better improve the overall data availability in the system.
In a system T(V, E), e(vi, vj), e(vk, vm) E, Ae(vi, vj)  Ae(vk, vm). The avaialiblity of such system
T(V, E) can be computed as ( , ) ( , )
( )* ( , )* ( ))) / ( )( ( A
r
ee x y v R v V
r
v V A v A x y N u A v    , where u  R and u(v, R).
Hence, the replica allocation problem to maximize the availability considering heterogeneity of
both node availability and network link availability can be defined as the following replica
allocation problem (Problem 1).
Problem 1: Given a tree network T(V, E), where V= {v1, v2, …,vn}, E is the set of all edges in T.
vi, vj V, NA(vi)  NA(vj), and e(vi, vj), e(vk, vm) E, Ae(vi, vj)  Ae(vk, vm). Find a subset of nodes
R  V and |R| = K that maximize the value of ( , ) ( , )
( )* ( , )* ( )),where ( , )( ee x y v R A
r
v V A v A x y N u v R Ru u
     .
In our previous work in [18], solutions have been given to availability optimization problem, by
considering the homogeneous of node availability or network link availability, or both. However,
these solutions are only partial solutions to Problem 1.
3.2 The Availability Enhancement Algorithm
Problem 1 is an optimization problem similar to the p-median problem [22, 15]. In a p-median
problem, it is given a graph G(V, E) where V= {v1, v2, …,vn}. Each node vi is associated with a
nonnegative weight ci, and each edge e(vi, vj) is associated with a nonnegative weight d(e(vi, vj)).
Additionally, it is given a real monotonically increasing function fi for each node vi. The goal is to
find a subset R  V containing at most p nodes (p is an arbitrary but pre-determined number) to
minimize the objective
(3)
Problem 1 and p-median problem have some common issues. They both consider weighted tree
networks where each node/edge is assigned value(s). They both look for optimal solutions to the
objective functions. Problem 1 looks for the maximum value of an objective function while p-
median problem looks for the minimum value of another objective function. It is therefore
possible for us to borrow some ideas of the solution to the p-median problem to develop the
solution to Problem 1.
However, it is difficult to directly apply the solution to the p-median problem to Problem 1 since
they are different. Consider the p-median problem. Suppose that a node vj is selected and the
edges e(vi, vi1), e(vi1, vi2), …, e(vik, vj) form a path connecting nodes vi and vj. Then the
computation of the cost for vi to access vj includes k+1 piece of information: fi(d(e(vi,vi1))),
fi1(d(e(vi1,vi2))), …, fik(d(e(vik,vj))), and cj. While for Problem 1, the computation of the
contribution for vi to fetch data from vj includes k+2 piece of information: Ar
(vi), Ae(vi,vi1),
Ae(vi1,vi2), …, Ae(vik,vj), and NA(vj). Therefore the solution to Problem 1 should include and handle
more information than that to p-median problem. Thus we need to modify the solution to p-
median problem to re-develop the solution to Problem 1.
Problem Transformation and Re-definition
A three step solution is provided to Problem 1:
1. Transformation: reduce Problem 1 from an arbitrary tree network to a binary tree network

9
2. Pre-processing: compute Aj
i
and rj
i
for each node vj, 1  i, j  n
3. Computing function values: compute the values of two functions G(vj,q,rj
i
) and F(vj,q,rj
i
)
based on their recursive relations.
Step 1. Transformation
Given an arbitrary tree network T(V, E), where V= {v1, v2, …,vn}, E is the set of all edges in T. It
can be transformed to that of a binary tree network T’(V’,E’). We show that by considering the
following two cases, Case 1 and Case 2.
Case1. Suppose that an internal node vj has exactly one child node (Fig. 5(a)). Without loss of
generality, we assume that vj has a left child node vj1. Then we introduce a right child node vj2 and
set the availabilities NA(vj2) = Ae(vj, vj2) = 0 and the number of read requests Ar
(vj2) = 0 (Fig.
5(b)).
Fig. 5 Node with a single child node (a) is transformed to (b)
Since NA(vj2) = Ae(vj, vj2) = Ar
(vj2) = 0, there is no benefit to select vj2 and add vj2 to the resident set
R. Therefore if R’ is an optimal solution to Problem 1 for T’(V’,E’), then R = R’ – {vj2 | j} is an
optimal solution to Problem 1 for T(V,E).
Case 2. Suppose that an internal node vj has more than 3 childe nodes vj1, vj2, ..., vjt where t  3
(Fig 6(a)). Then we introduct t-2 nodes ujs where 2  s  t − 1, replace edge e(vj, vjs) by edge e(ujs,
vjs) for 2  s  t − 1, replace edge e(vj, vjt) by edge e(ujt−1, vjt), and add edges e(vj, uj2) and e(ujs’,
ujs’+1) for 2  s’  t – 2, and set the availabilities NA(ujs) = 0 for 2  s  t − 1, Ae(ujs, vjs) = Ae(vj, vjs)
for 2  s  t − 1, Ae(ujt−1, vjt) = Ae(vj, vjt), Ae(vj, uj2) = Ae(ujs’, ujs’+1) = 1 for 2  s’  t – 2, and the
number of read requests Ar
(ujs) = 0 for 2  s  t – 1 (Fig 6 (b)).
Fig. 6 (a). Case 2, the node with over 3 child nodes.

10
Fig. 6 (b). Transformed graph for the graph shown in Fig. 6 (a).
First, it can be verified that the success availability for node vj (vjs) to fetch data from node vjs (vj)
does not change, 2  s  t – 1. Second, the new assigned availabilities and number of requests
guarantee that there is no benefit to select nodes ujs, 2  s  t − 1. Therefore if R’ is an optimal
solution to Problem 1 for T’(V’,E’), then R = R’ – {ujs | j, s} is an optimal solution to Problem 1
for T(V,E).
Based on the previous discussion, Problem 1 can be efficiently transformed from an arbitrary tree
network to a binary tree network. The solution to the transformed problem can be easily
transformed back to the original problem.
Step 2. Pre-processing:
In this step we compute Aj
i
and the corresponding rj
i
for each node vj, 1  i, j  n. Aj
i
and rj
i
are
defined in the following definition.
Definition 1: Aj
1
is the greatest success probability for node vj to fetch data from all nodes, …, Aj
n
is the smallest success probability for node vj to fetch data from all nodes. rj
1
is the node
corresponds to Aj
1
, …, rj
n
is the node corresponds to Aj
n
.
Given a binary tree network, we compute the success probability pij for node vi to fetch data from
node vj. Specifically, if the edges e(vi, vi1), e(vi1, vi2), …, e(vik, vj) form the path connecting vi and
vj, then the probability pij = Ae(vi,vi1)Ae(vi1,vi2) … Ae(vik,vj) NA(vj). Then we sort p1j, …, pnj from
the smallest to the largest, and let the results be Aj
1
, …, Aj
n
. Additionally, let node rj
i
corresponds
to Aj
i
, 1  i  n. The equality of success probabilities can be resolved by arbitrary set the order of
nodes.
Step 3. Computing recursive function values
Let vj be a node, q be an integer, and rj
i
and Aj
i
be the values computed in the pre-processing step.
The functions G(vj,q,rj
i
) and F(vj,q,rj
i
) are defined in the following definition.
Definition 2: G(vj,q,rj
i
) is defined to be the optimal value of the sub-problem defined on the sub-
tree Tj given that total of at least 1 and at most q nodes can be selected in Tj, and at least one node
1
, …, rj
i
} ∩ Vj. The set of nodes for G(vj,q,rj
i
) to achieve the optimal value
is called the associated set of G(vj,q,rj
i
).
Definition 3: F(vj,q,rj
i
) is defined to be the optimal value of the sub-problem defined on the sub-
tree Tj under the constraints that a total of at most q nodes can be selected in Tj and there are

11
already some selected nodes in V  Vj, and the best amongst them that vj can fetch data is rj
i
. The
set of nodes for F(vj,q,rj
i
) to achieve the optimal value is called the associated set of F(vj,q,rj
i
).
We establish the recursive relations between G(vj,q,rj
i
) and F(vj,q,rj
i
) as what follows.
Leaf node vi
For function G(vj,q,rj
i
),
G(vj,q,rj
i
)=∞ if vj{rj
1
, …, rj
i
} or q  0
G(vj,q,rj
i
)=Ar
(vj)Aj
s
if vj=rj
s
{rj
1
, …, rj
i
} and q  1
For function F(vj,q,rj
i
),
F(vj,q,rj
i
)=∞ if rj
i
Vj or q < 0
F(vj,q,rj
i
)=Ar
(vj)Aj
i
if rj
i
Vj and q = 0
F(vj,q,rj
i
)=max{F(vj,0,rj
i
),G(vj,1,vj)} if rj
i
Vj and q  1
Internal node vi
For function G(vj,q,rj
i
),
G(vj,q,rj
i
)=∞ if Vj∩{rj
1
, …, rj
i
}= or q  0 or i = 0
G(vj,q,rj
i
)=G(vj,q,rj
i1
) if rj
i
Vj
G(vj,q,rj
i
) = max{G(vj,q,rj
i1
),Ar
(vj)Aj
i
+ max{F(vj1,q1,rj1
i1
)+F(vj2,q2,rj2
i2
) | 0q1,q2q1,
q1+q2=q1}}
if rj
i
=vj and q  1, rj
i
corresponds to rj1
i1
for vj1 and rj2
i2
for vj2
G(vj,q,rj
i
)=max{G(vj,q,rj
i1
),Ar
(vj)Aj
i
+max{G(vj1,q1,rj1
i1
)+F(vj2,q2,rj2
i2
) | 1q1q, 0q2q,
q1+q2=q, the associated set of G(vj1,q1,rj1
i1
1
, …, rj
i1
but contains rj
i
}}
if rj
i
Vj1 and q  1, rj
i
corresponds to rj1
i1
for vj1 and rj2
i2
for vj2
G(vj,q,rj
i
)=max{G(vj,q,rj
i1
),Ar
(vj)Aj
i
+max{F(vj1,q1,rj1
i1
)+G(vj2,q2,rj2
i2
) | 0q1q, 1q2q,
q1+q2=q, the associated set of G(vj2,q2,rj2
i2
1
, …, rj
i1
but contains rj
i
}}
if rj
i
Vj2 and q  1, rj
i
corresponds to rj1
i1
for vj1 and rj2
i2
for vj2
For function F(vj,q,rj
i
),
F(vj,q,rj
i
)=∞ if rj
i
Vj or q < 0
F(vj,q,rj
i
)=max{G(vj,q,rj
i
),Ar
(vj)Aj
i
+max{F(vj1,q1,rj1
i1
)+F(vj2,q2,rj2
i2
) | 0q1q, 0q2q,
q1+q2=q}}
if rj
i
Vj and q  0, rj
i
corresponds to rj1
i1
for vj1 and rj2
i2
for vj2
The Algorithm for the Transformed Problem
We integrate the three steps described in previous subsections to develop the algorithm,
Alg_Enhan, to solve the transformed Problem 1.
1.transforms the problem to a binary tree network if necessary,
2.pre-processes the binary tree network and compute Aj
i
and rj
i
for each node vj, 1  i, j  n,
3.computes the values of G(vj,q,rj
i
) and F(vj,q,rj
i
) based on the recursive relations following the
post order of the nodes: first compute the function values for the leaf nodes, then the internal
nodes, and finally the root node. The associated sets of G(vj,q,rj
i
) and F(vj,q,rj
i
) are recorded
at the same time. If two sets of nodes both achieve the optimal value, then the set containing
the node in Vj from which it is better for vj to fetch data will be recorded, and
4.Returns G(v1,K,r1
n
) together with the associated set, where v1 is the root node.

12
3.3 A Numerical Example for the Availability Enhancement Algorithm (Alg_Enhan)
We now use a numerical example to demonstrate the algorithm we developed for Problem 1.
Consider the binary tree network T(V, E) as shown in fig. 7, where V = {v1, v2, v3, v4, v5} and E =
{e(v1, v2), e(v1, v3), e(v2, v4), e(v2, v5)}. The nodes availabilities NA(v1) = 0.3, NA(v2) = 0.1, NA(v3) =
0.6, NA(v4) = 0.7, NA(v5) = 0.9. The edges availabilities Ae(v1, v2) = 0.5, Ae(v1, v3) = 0.4, Ae(v2, v4) =
0.2, Ae(v2, v5) = 0.8. The numbers of read requests Ar
(v1) = 100, Ar
(v2) = 200, Ar
(v3) = 300, Ar
(v4)
= 400, Ar
(v5) = 500. The tree network is illustrated in the following figure. We further assume that
K = 2.
Fig. 7. A sample network for algorithm numerical demonstration.
Pre-processing:
Since it is a binary tree network, transformation is not needed. We compute Aj
i
and rj
i
in this
subsection.
Root node v1
The success probability to fetch data from v1 = 0.3.
The success probability to fetch data from v2 = 0.50.1 = 0.05.
The success probability to fetch data from v4 = 0.50.20.7 = 0.07.
We sort the success probabilities and record A1
i
and the corresponding r1
i
in the following table,
1i5.
Table 1. Pre-processing the data access for root node v1
r1
1
=v5 r1
2
=v1 r1
3
=v3 r1
4
=v4 r1
5
=v2
A1
1
=0.36 A1
2
=0.3 A1
3
=0.24 A1
4
=0.07 A1
5
=0.05
Internal node v2

13
i
i
1i5.
Table 2. Pre-processing the data access for internal node v2
r2
1
=v5 r2
2
=v1 r2
3
=v4 r2
4
=v3 r2
5
=v2
A2
1
=0.72 A2
2
=0.15 A2
3
=0.14 A2
4
=0.12 A2
5
=0.1
Leaf node v3
The success probability to fetch data from v4 = 0.40.50.20.7 = 0.028.
i
i
1i5.
Table 3.Pre-processing the data access for leaf node v3
r3
1
=v3 r3
2
=v5 r3
3
=v1 r3
4
=v4 r3
5
=v2
A3
1
=0.6 A3
2
=0.144 A3
3
=0.12 A3
4
=0.028 A3
5
=0.02
Leaf node v4
i
i
1i5.
Table 4.Pre-processing the data access for leaf node v4
r2
1
=v5 r2
2
=v1 r2
3
=v4 r2
4
=v3 r2
5
=v2
A2
1
=0.72 A2
2
=0.15 A2
3
=0.14 A2
4
=0.12 A2
5
=0.1
Leaf node v5
i
i
1i5.

14
Table 5. Pre-processing the data access for leaf node v5
r5
1
=v5 r5
2
=v1 r5
3
=v4 r5
4
=v3 r5
5
=v2
A5
1
=0.9 A5
2
=0.12 A5
3
=0.112 A5
4
=0.096 A5
5
=0.08
Recursive function computing:
We now compute the function values based on the pre-processing results and recursive relations
developed above. We follow the post order of the nodes, which is v4 → v5 → v2 → v3 → v1.
Leaf node v4
The G and F function values for node v4 is summarized in the Table 6. The associated set of nodes
is also recorded.
Table 6. G and F function values for node v4
G(v4,1,r4
1
)=280;
v4
G(v4,2,r4
1
)=280;
v4
F(v4,0,r4
1
)=∞ F(v4,1,r4
1
)=∞ F(v4,2,r4
1
)=∞
G(v4,1,r4
2
)=280;
v4
G(v4,2,r4
2
)=280;
v4
F(v4,0,r4
2
)=57.6 F(v4,1,r4
2
)=280;
v4
F(v4,2,r4
2
)=280;
v4
G(v4,1,r4
3
)=280;
v4
G(v4,2,r4
3
)=280;
v4
F(v4,0,r4
3
)=12 F(v4,1,r4
3
)=280;
v4
F(v4,2,r4
3
)=280;
v4
G(v4,1,r4
4
)=280;
v4
G(v4,2,r4
4
)=280;
v4
F(v4,0,r4
4
)=9.6 F(v4,1,r4
4
)=280;
v4
F(v4,2,r4
4
)=280;
v4
G(v4,1,r4
5
)=280;
v4
G(v4,2,r4
5
)=280;
v4
F(v4,0,r4
5
)=8 F(v4,1,r4
5
)=280;
v4
F(v4,2,r4
5
)=280;
v4
Leaf node v5
is also recorded.
Table 7.G and F function values for node v5
G(v5,1,r5
1
)=450;
v5
G(v5,2,r5
1
)=450;
v5
F(v5,0,r5
1
)=∞ F(v5,1,r5
1
)=∞ F(v5,2,r5
1
)=∞
G(v5,1,r5
2
)=450;
v5
G(v5,2,r5
2
)=450;
v5
F(v5,0,r5
2
)=60 F(v5,1,r5
2
)=450;
v5
F(v5,2,r5
2
)=450;
v5
G(v5,1,r5
3
)=450;
v5
G(v5,2,r5
3
)=450;
v5
F(v5,0,r5
3
)=56 F(v5,1,r5
3
)=450;
v5
F(v5,2,r5
3
)=450;
v5
G(v5,1,r5
4
)=450;
v5
G(v5,2,r5
4
)=450;
v5
F(v5,0,r5
4
)=48 F(v5,1,r5
4
)=450;
v5
F(v5,2,r5
4
)=450;
v5
G(v5,1,r5
5
)=450;
v5
G(v5,2,r5
5
)=450;
v5
F(v5,0,r5
5
)=40 F(v5,1,r5
5
)=450;
v5
F(v5,2,r5
5
)=450;
v5
Internal node v2
is also recorded.

15
Table 8. G and F function values for node v2
G(v2,1,r2
1
)=651.6;
v5
G(v2,2,r2
1
)=874;
v4,v5
F(v2,0,r2
1
)=∞ F(v2,1,r2
1
)=∞ F(v2,2,r2
1
)=∞
G(v2,1,r2
2
)=651.6;
v5
G(v2,2,r2
2
)=874;
v4,v5
F(v2,0,r2
2
)=102 F(v2,1,r2
2
)=651.6;
v5
F(v2,2,r2
2
)=874;
v4,v5
G(v2,1,r2
3
)=651.6;
v5
G(v2,2,r2
3
)=874;
v4,v5
F(v2,0,r2
3
)=∞ F(v2,1,r2
3
)=∞ F(v2,2,r2
3
)=∞
G(v2,1,r2
4
)=651.6;
v5
G(v2,2,r2
4
)=874;
v4,v5
F(v2,0,r2
4
)=81.6 F(v2,1,r2
4
)=651.6;
v5
F(v2,2,r2
4
)=874;
v4,v5
G(v2,1,r2
5
)=651.6;
v5
G(v2,2,r2
5
)=874;
v4,v5
F(v2,0,r2
5
)=∞ F(v2,1,r2
5
)=∞ F(v2,2,r2
5
)=∞
Leaf node v3
is also recorded.
Table 9. G and F function values for leaf node v3
G(v3,1,r3
1
)=180; v3 G(v3,2,r3
1
)=180;
v3
F(v3,0,r3
1
)=∞ F(v3,1,r3
1
)=∞ F(v3,2,r3
1
)=∞
G(v3,1,r3
2
)=180; v3 G(v3,2,r3
2
)=180;
v3
F(v3,0,r3
2
)=43.2 F(v3,1,r3
2
)=180;
v3
F(v3,2,r3
2
)=180;
v3
G(v3,1,r3
3
)=180; v3 G(v3,2,r3
3
)=180;
v3
F(v3,0,r3
3
)=36 F(v3,1,r3
3
)=180;
v3
F(v3,2,r3
3
)=180;
v3
G(v3,1,r3
4
)=180; v3 G(v3,2,r3
4
)=180;
v3
F(v3,0,r3
4
)=8.4 F(v3,1,r3
4
)=180;
v3
F(v3,2,r3
4
)=180;
v3
G(v3,1,r3
5
)=180; v3 G(v3,2,r3
5
)=180;
v3
F(v3,0,r3
5
)=6 F(v3,1,r3
5
)=180;
v3
F(v3,2,r3
5
)=180;
v3
Root node v1
The G and F function values for node v1 is summarized in the Table 10. The associated set of
nodes is also recorded.
Table 10.G and F function values for root node v1
G(v1,1,r1
1
)=730.8;
v5
G(v1,2,r1
1
)=953.2;
v4,v5
F(v1,0,r1
1
)=∞ F(v1,1,r1
1
)=∞ F(v1,2,r1
1
)=∞
G(v1,1,r1
2
)=730.8;
v5
G(v1,2,r1
2
)=953.2;
v4,v5
F(v1,0,r1
2
)=∞ F(v1,1,r1
2
)=∞ F(v1,2,r1
2
)=∞
G(v1,1,r1
3
)=730.8;
v5
G(v1,2,r1
3
)=953.2;
v4,v5
F(v1,0,r1
3
)=∞ F(v1,1,r1
3
)=∞ F(v1,2,r1
3
)=∞
G(v1,1,r1
4
)=730.8;
v5
G(v1,2,r1
4
)=953.2;
v4,v5
F(v1,0,r1
4
)=∞ F(v1,1,r1
4
)=∞ F(v1,2,r1
4
)=∞
G(v1,1,r1
5
)=730.8;
v5
G(v1,2,r1
5
)=953.2;
v4,v5
F(v1,0,r1
5
)=∞ F(v1,1,r1
5
)=∞ F(v1,2,r1
5
)=∞
According to the entry (G(v1,2,r1
5
) = 953.2; v4,v5) in Table 10 function values for root node v1,
when selecting nodes v4 and v5, the objective function in Problem 2 achieves the maximum value
953.2.

16
3.4 Complexity and Correctness Analysis for the Availability Enhancement Algorithm
(Alg_Enhan)
Complexity Analysis for the Algorithm (Alg_Enhan)
We now analyze the runtime complexity for the availability enhancement algorithm (Alg_Enhan).
For simplicity, we assume that the computational complexity for arithmetic operations (+, , , /)
and comparison is O(1).
Theorem 1: The computational complexity of the algorithm is O(Kn2
).
Proof. In step 1 (transformation), O(n) nodes and edges may be added to the original tree
network. The corresponding parameters (e.g. probabilities and number of read requests) may also
be added. So the computational complexity of step 1 is O(n).
In step 2 (pre-processing), the success probabilities for each node to fetch data from all nodes are
computed and sorted. It is possible that the computation of a success probability costs O(n)
multiplications (consider unbalanced binary tree). So the computational complexity of step 2 is
O(n)(O(n)O(n)+O(nlogn)) = O(n3
). However, the computational complexity can be improved to
O(n2
) based on the similar technique used in [25, 15].
In step 3 (computing function values), the values of two functions are computed based on the
recursive relations. O(n) tables are computed for O(n) nodes. Each table contains O(n)O(K) =
O(Kn) entries. In the worst case, the computation of an entry requires O(K) operations. So the
computational complexity of step 3 is O(n)O(Kn)O(K) = O(K2
n2
). However, the computational
complexity can be improved to O(Kn2
) based on the similar technique used in [26, 15].
Therefore the computational complexity of the algorithm is O(n) + O(n2
) + O(Kn2
) = O(Kn2
) 
Correctness Analysis for the Algorithm(Alg_Enhan)
We now show the correctness in Theorem 2 for the availability enhancement algorithm
(Alg_Enhan).
Theorem 2: The algorithm Alg_Enhan outputs the optimal solution of Problem 1.
Proof. It suffices to prove the correctness of recursive relations.
Let vj be a leaf node. If vj{rj
1
, …, rj
i
} or q  0, the value of G(vj,q,rj
i
) should not be counted
since the situation is not included in the definition. Therefore we set G(vj,q,rj
i
)=∞. On the
contrary, if vj=rj
s
{rj
1
, …, rj
i
} and q  1, G(vj,q,rj
i
) equals to the optimal value of the sub-problem
defined on the sub-tree Tj, which is the node vj. Therefore G(vj,q,rj
i
) = Ar
(vj)Aj
s
.
If rj
i
i
i
)=∞. If rj
i
Vj and q = 0, the leaf node vj has to fetch
data from node rj
i
. Therefore F(vj,q,rj
i
)=Ar
(vj)Aj
i
. If rj
i
Vj and q  1, the leaf node vj can either
fetch data from node rj
i
or fetch data from itself. Therefore F(vj,q,rj
i
)=max{F(vj,0,rj
i
),G(vj,1,vj)}.
Let vj be an internal node. If Vj∩{rj
1
, …, rj
i
}= or q  0 or i = 0, the value of G(vj,q,rj
i
) should not
be counted since the situation is not included in the definition. Therefore we set G(vj,q,rj
i
)=∞. If
rj
i
Vj, at least one node must be selected in {rj
1
, …, rj
i
} ∩ Vj is equivalent to at least one node
1
, …, rj
i1
} ∩ Vj. Therefore G(vj,q,rj
i
)=G(vj,q,rj
i1
). The rest situations can be

17
separated into 3 cases: rj
i
=vj and q  1, rj
i
Vj1 and q  1, and rj
i
Vj2 and q  1. We discuss them as
follows.
In case 1 we consider that rj
i
=vj and q  1. Without loss of generality, we assume that rj
i
corresponds to rj1
i1
i2
for vj2 (the right child node of vj).We
claim that
G(vj,q,rj
i
)=max{G(vj,q,rj
i1
),Ar
(vj)Aj
i
+max{F(vj1,q1,rj1
i1
)+F(vj2,q2,rj2
i2
)| 0q1,q2q1, q1+q2=q1}}.
In order to prove that, we must show (1) the optimal solution appears either in G(vj,q,rj
i1
) or
Ar
(vj)Aj
i
+max{F(vj1,q1,rj1
i1
)+F(vj2,q2,rj2
i2
) | 0q1,q2q1, q1+q2=q1}, and (2) the optimal solution
is the maximum value.
1
, …, rj
i1
i1
1
, …, rj
i1
i
is selected in the optimal solution. Then it is better for vj to fetch data from node rj
i
=vj than other
nodes. Let q1 nodes will be selected from Tj1 and q2 nodes will be selected from Tj2 where
q1+q2=q1. Hence the optimal solution appears in Ar
(vj)Aj
i
+max{F(vj1,q1,rj1
i1
)+F(vj2,q2,rj2
i2
) |
0q1,q2q1, q1+q2=q1}.
Second, Ar
(vj)Aj
i
+max{F(vj1,q1,rj1
i1
)+F(vj2,q2,rj2
i2
) | 0q1,q2q1, q1+q2=q1} may contain some
inconsistent values. For example, if F(vj1,q1,rj1
i1
) or F(vj2,q2,rj2
i2
) selects a node in {rj
1
, …, rj
i1
} ∩
Vj, vj should fetch data from that node instead of rj
i
. However since the term Ar
(vj)Aj
i
is used, it
actually counts that vj fetches data from node rj
i
. Or F(vj1,q1,rj1
i1
) selects a node such that vj2
should fetch data from that node outside Tj2 instead of rj2
i2
. However the term F(vj2,q2,rj2
i2
)
indicates that actually it counts that vj2 fetches data from node rj2
i2
outside Tj2. Fortunately, the
inconsistent values are smaller than the corresponding consistent values, and hence are smaller
than the optimal solution. Therefore the inconsistent values will not affect the optimal solution to
be the maximum value.
i
i
corresponds
to rj1
i1
i2
for vj2 (the right child node of vj). We claim that
G(vj,q,rj
i
)=max{G(vj,q,rj
i1
),Ar
(vj)Aj
i
+max{G(vj1,q1,rj1
i1
)+F(vj2,q2,rj2
i2
) | 1q1q, 0q2q, q1+q2=q,
i1
1
, …, rj
i1
but contains rj
i
}}.
1
, …, rj
i1
i1
1
, …, rj
i1
i
is selected in the optimal solution. We claim that vj will not be selected. Because if vj is selected,
then it cannot be in {rj
1
, …, rj
i1
} ∩ Vj. Thus it is better for vj to fetch data from node rj
i
than vj.
Consequently it is better for all nodes to fetch data from node rj
i
than vj. It implies that the
selection of vj will not gain any benefit. Therefore vj will not be selected. So q1 nodes will be
selected from Tj1 and q2 nodes will be selected from Tj2 where q1+q2=q. Hence the optimal
solution appears in Ar
(vj)Aj
i
+max{G(vj1,q1,rj1
i1
)+F(vj2,q2,rj2
i2
) | 1q1q, 0q2q, q1+q2=q, the
associated set of G(vj1,q1,rj1
i1
1
, …, rj
i1
but contains rj
i
}.
Second, if the associated set of G(vj1,q1,rj1
i1
1
, …, rj
i
, then that G(vj1,q1,rj1
i1
)
should not be counted. If the associated set of G(vj1,q1,rj1
i1
) contains rj
1
, …, rj
i1
, then the
inconsistent value is smaller than the corresponding consistent value, and hence smaller than the
optimal value. so that G(vj1,q1,rj1
i1
) can be omitted. Therefore it suffices to consider G(vj1,q1,rj1
i1
)
whose associated set does not contain rj
1
, …, rj
i1
but contains rj
i
.

18
i
i
corresponds
to rj1
i1
i2
for vj2 (the right child node of vj). Analogous to the
proof of case 2, it can be proved that
G(vj,q,rj
i
)=max{G(vj,q,rj
i1
),Ar
(vj)Aj
i
+max{F(vj1,q1,rj1
i1
)+G(vj2,q2,rj2
i2
) | 0q1q, 1q2q, q1+q2=q,
i2
1
, …, rj
i1
but contains rj
i
}}.
If rj
i
i
i
)=∞. If rj
i
Vj and q  0, without loss of generality,
we assume that rj
i
corresponds to rj1
i1
i2
for vj2 (the right
child node of vj). We claim that
F(vj,q,rj
i
)=max{G(vj,q,rj
i
),Ar
(vj)Aj
i
+max{F(vj1,q1,rj1
i1
)+F(vj2,q2,rj2
i2
) | 0q1q, 0q2q, q1+q2=q}}
First, if the optimal solution selects node in {rj
1
, …, rj
i
} ∩ Vj, then the optimal solution appears in
G(vj,q,rj
i
).
Otherwise we assume that no nodes in {rj
1
, …, rj
i
} ∩ Vj is selected in the optimal solution. Then it
is better for vj to fetch data from node rj
i
than other nodes. Additionally vj will not be selected
because of the same reason in the proof of case 2. So q1 nodes will be selected from Tj1 and q2
nodes will be selected from Tj2 where q1+q2=q. Hence the optimal solution appears in
Ar
(vj)Aj
i
+ max{F(vj1,q1,rj1
i1
)+F(vj2,q2,rj2
i2
) | 0q1q, 0q2q, q1+q2=q}.
Finally, the inconsistent values in Ar
(vj)Aj
i
+max{F(vj1,q1,rj1
i1
)+F(vj2,q2,rj2
i2
) | 0q1q, 0q2q,
q1+q2=q} are smaller than the corresponding consistent values, and hence are smaller than the
optimal solution. Therefore the inconsistent values will not affect the optimal solution to be the
maximum value.
4. EXPERIMENTAL STUDIES
4.1 Evaluating the Effectiveness of Availability Modeling in Tree Networks
In section II, we have shown that the proposed availability modeling algorithm in a tree network
has a complexity of |V|2
, where |V| is the number of nodes in the tree network. In this section, we
investigate the practical complexity of our proposed algorithm. In this simulation, a variety of
random tree topologies are generated. We use four parameters to control the generation of a tree:
the total number of nodes (|V|), resident size (|R|), the number of reads issued from each node
(Ar
(u)), and the maximum node degree (degree). A random tree is generated in a breadth-first
manner, i.e., starting from the root node, at each level, a random number of node are generated for
the next level based on the randomly generated node degrees, until the number of nodes specified
is reached. The root of the tree always hosts a replica and the rest of |R|-1 replicas are allocated to
randomly selected |R|-1 nodes. The number read on each node is randomly generated following a
uniform distribution, each in the range of (0 10). Link availability (LA) of each edge and node
availability (NA) of each node in the system is generated randomly following a uniform
distribution, each is in the range of (0.9 1.0).
Table 11. Performance of the Algorithm in a tree
Node
Number
Memory
(KB) Runtime
100 1164 15 ms
1000 4787 1.34s
10000 7952 1.84 min.
100000 28922 2.06 hr

19
To study actual performance of the algorithm, the actual runtime and memory used are
determined. The memory used is dynamic and the maximum memory used is measured in the
middle of the recursive algorithm execution process. The runtime is determined as the time
between the start of the execution of the algorithm and end of the algorithm. Note that the actual
runtime and memory used here do not necessarily mean the lowest runtime and maximum
memory used, and the algorithm is implemented in Java. The parameters are set as follows:
degree = 5, each node can have 0 to 5 children, |R| = 6, and |V| is ranged from 100 to 100000. As
can be seen in Table 11, the memory used by the algorithm linearly increased along the increase
of |V|, while the run time of the algorithm increases along with the increase of |V|, governed by a
growth function of the square of |V|. The results confirmed our theoretic analysis in Section II
such that the availability modeling algorithm in a tree network has a runtime complexity of |V|2
.
We then test our modeling approach against the simulated distributed environment to see how
good it is. The impacts of three factors on Tree_SA(T, R) are studied: (a) the graph size |V|; (b) the
graph degree, which is the average number of neighbors of a node; and (c) the size of the resident
set |R|. To avoid biased access patterns and topology structures, we repeat the experimental steps
100 times. The final result is the average of the 100 trials. The results are shown in Fig. 8 (a), Fig.
8 (b), and Fig. 8 (c).
Fig. 8 (a). The impact of resident set size
Fig.8(a) shows the impact of resident set size on the effectiveness of the three replica allocation
algorithms. The parameters are set as follows: |V| = 100, degree = 5, and |R| is increased from 1 to
20. The result shows that data availabilities of the three replica allocation schemes increase along
with the increase of |R|, which confirms the conclusion of Theorem 1. From Fig. 7(a), the two
resident sets computed by the two replica allocation algorithms, Alg_Hom and Alg_Het, can
always achieve higher availabilities than the availability achieved by the resident set that is
computed by the randomized replication allocation algorithm. However, the difference on the
availabilities of the resident sets between the randomized and the two algorithms, Alg_Hom and
Alg_Het, decreases along with increase of |R|. Also, the availability increase rate decreases along
with increase of |R|. The reason is obvious. With a larger |R|, the average distance from a user to
service/data site (number of edges in the path) will be reduced and thus the service/data reach
availability should be increased. Thus, the space of service availability enhancement will also be
reduced when availability reaches a high level.

20
Fig. 8 (b). The impact of node degree
Fig.8(b) shows the impact of node degree on the effectiveness of the three replica allocation
algorithms. The parameters are set as follows: |V| = 100, |R| = 6, and degree is increased from 2 to
10. The result shows that data availabilities of the three replica allocation schemes increase along
with the increase of node degree. From Fig. 7(b), the two resident sets computed by the two
replica allocation algorithms, Alg_Hom and Alg_Het, can always achieve higher availabilities
than the availability achieved by the resident set that is computed by the randomized replication
allocation algorithm. However, the difference on the availabilities of the resident sets between the
randomized and the two algorithms, Alg_Hom and Alg_Het, decreases along with increase of
node degree. The availabilities increase rate also slows down when node degree reach a certain
size (e.g., degree = 5). The reason is obvious. With a larger node degree, the height of the tree
will be reduced. Therefore, the average distance from a user to service/data site (number of edges
in the path) will be reduced and thus the service/data reach availability should be increased. Thus,
the space of service availability enhancement will also be reduced when availability reaches a
high level.
0.8
0.82
0.84
0.86
0.88
0.9
0.92
0 100 200 300 400 500 600 700 800 900 1000
Number of Node in the System
Availability
Tree_SA(T, R)
Fig. 8 (c). The impact of graph size on data availability
Fig.8(c) shows the impact of graph size on the data availability modeled and computed by
Tree_SA(T, R). The parameters are set as follows: |R| = 6, degree = 5 and |V| is increased from 100
to 1000. The result shows that data availability decreases along with increase of graph size. The
reason is obvious. With a larger |V|, the average distance from a user to service/data site (number
of edges in the path) will be increased and thus the service/data reach availability should
decrease.
4.2 Evaluating the Effectiveness of Availability Enhancement Algorithms in Tree
Networks
In this section, we test two replica allocation algorithms we developed in this research in a
simulated distributed environment to see how effective they are on enhancing service
availabilities. Specifically, we compare the availability enhanced algorithm, Alg_Enhan, with the

21
randomized replica allocation scheme, Alg_Ran. In this simulation, a variety of random tree
networks are generated. We use to simulate a tree with 30 nodes which have a variety of resident
size (|R. A random tree is generated in a breadth-first manner, i.e., starting from the root node, at
each level, a random number of the node are generated for the next level based on the randomly
generated node degrees, until the number of nodes specified is reached. The entire tree hosts |R|
replicas. The number read on each node is randomly generated following a uniform distribution,
each is in the range of (0, 10). Link availabilities of all the edges and node availabilities of all the
nodes in the system are generated randomly following a uniform distribution in the range of (0.9,
1.0). For the randomized allocation scheme, replicas are allocated to the randomly selected |R|
nodes in the system. For algorithms Alg_Enhan, we will first implement the algorithm based on a
tree network, and then use this algorithm to compute the resident sets in the simulated tree
network. Finally, we will apply our availability computing algorithm shown in Fig. 3 to compute
the service availabilities for the two resident sets in the simulated tree network.
Our goal is to test which replica allocation scheme is more effective in enhancing overall service
availability in a tree network. The impacts of the resident set size |R| (from 1 to 6). The
experiments are repeated 10 times and the results are shown in Fig. 9.
Fig.9. The impact of resident set size on improved availability
Fig.9 shows the impact of resident set size on the effectiveness of the three replica allocation
algorithms. The parameters are set as follows: |V| = 30, degree = 5, and |R| is increased from 1 to
6. The result shows that data availabilities of the two replica allocation schemes increase along
with the increase of |R|, which confirms the conclusion of Theorem 1. From Fig. 8, the resident
set computed by Alg_Enhan can always achieve higher availabilities than the availability
achieved by the resident set that is computed by Alg_Ran. However, the difference on the
availabilities of the resident sets between Alg_Ran and Alg_Enhan decreases along the increase of
|R| (from 0.05 to 0.002). Also, the availability increase rate decreases along the increase of |R|.
The reason is obvious. With a larger |R|, the average distance from a user to service/data site
(number of edges in the path) will be reduced and thus the service/data reach availability should
be increased. Thus, the space of service availability enhancement will also be reduced when
availability reaches a high level. From the above experimental study, we can conclude that when
the replica size is small, the replica location has the bigger impact on availability. Hence, it is
more critical to employ a better replica allocation algorithm to improve service availability, such
as Alg_Enhan.

22
5. RELATED WORK
Network reliability has been well studied and a large amount of works have been conducted on
this topic [15, 23, 24, 31]. The modeling and computation of network reliability is similar to (but
not the same as) a specific case of network availability modeling for a distributed system, i.e.,
modeling the availability of a distributed system with no replication. For such a system, the data
availability can be computed by computing the reach availability from each node to the
destination node (the node with the data). The difference between availability and reliability
modeling is that the unavailability of a node does not affect the reaching probability in our data
availability modeling (in reliability, the node unavailability will affect the reliability). However,
the availability modeling in a distributed system with data replication is very different from
reliability modeling. The data availability in the system to a node w (actually the clients
associated with such node) is affected by multiple nodes (each with a copy of the data) at the
same time, i.e., if a copy of data on node u is not available to node w, then another node v can
provide availability to w. Therefore, computing and modeling data availability to node w not only
needs to compute the reach probability to multiple destination nodes (for example u and v), but
also needs to consider their impact on each other.
Availability has long been considered as a critical issue for systems providing replicated services.
In [8, 9], the authors explored the benefits of dynamically trading consistency for availability
using a continuous consistency model. The availability model is a function of consistency,
workload, and fault load. If consistency requirement is greatly relaxed, availability can be very
high. In data storage systems such as Ocean store [14, 27] and PASIS [28], both pure replication
and data fragmentation approaches are considered. Their availability models only consider node
availability. [11] and [12] both present an analytical model for determining the overall availability
of data stored in a data grid or P2P system. The availability model captures the characteristics of
peer-to-peer systems and grid systems. The availability of replica location service (metadata
service) of data grids or P2P systems can be considered independent of data storage availability.
All the above works on data availability computation only consider the availability of the service
or data sites and do not consider network link availability. In [6, 7], the authors propose a model
for studying the availability of replicated systems. The system availability is computed in a
stochastic graph, i.e. each node and each link are assigned a specific failure probability. The
overall system availability is computed exhaustively using a state enumeration method. However,
the runtime complexity of the state enumeration method is exponential in the number of nodes in
the system in a tree network. Also, it does not give a solution to compute data availability with
multiple replicas in the system. In our previous research in [17, 29], we proposed effective
availability modeling and computing approaches that consider both node and network link
failures, for systems with the tree, ring, and general graph topologies. In [18], we provided two
availability enhancement algorithms for a tree network with homogeneous node availability and
link availability, which is a special case for the problem we consider in this research. Hence, a
solution is needed for the general case, i.e., each node availability and link (network edge)
availability are heterogeneous.
6. CONCLUSIONS
In this paper, we address the system service or data availability modeling and enhancement
approaches for distributed systems with data replication. We first provide an efficient availability
computing algorithm for tree networks. We then analyze the availability enhancement problem
and formalized availability enhancement in a general tree network as Problem 1. Then, the
problems are transformed into a special p-median problem and a recursive algorithm-based
solution, Alg_Enhan is developed to maximize the service availabilities for distributed systems
with a tree topology. The algorithm Alg_Enhan considers the impact of replica location on overall

23
service availability and the impact of network node and edge heterogeneity. Theorems have
shown that the algorithm Alg_Enhan is efficient with a runtime complexity of O(K|V|2), where K
is the number of replicas and |V| is the number of nodes in the network. Finally, experimental
studies show that the algorithm Alg_Enhan can compute resident sets with higher availabilities
than those computed by the randomized algorithm, but it is more effective when the number of
replicas is small.
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25
AUTHORS
Michael Tu is a Ph.D. in Computer Science, Associate Professor of Computer
Information Technology, director of the Center for Cyber Security and Infrastructure
Protection at Purdue University Northwest. Dr. Tu’s areas of expertise are information
assurance, digital forensics, and cybersecurity education. He has published over 40 peer
reviewed papers in prestigious journals and peer reviewed conference proceedings. Dr.
Tu is a Certified Information System Security Professional (CISSP), Certified Ethical
Hacker (CEH), Certified Hacking and Forensics Investigator (CHFI) and AccessData
Computer Examiner (ACE).
Dianxiang Xu, received the B.S., M.S., and Ph.D. degrees in computer science from
Nanjing University, Nanjing, China. He is a Professor with the Department of Computer
Science, Boise State University, Boise, ID, USA. His research interests include software
security and safety, software engineering, access control, and software-defined systems.
Dr. Xu has published over 100 peer reviewed papers in prestigious journals and peer
reviewed conference proceedings.
Liangliang Xiao, received the M.S., and Ph.D. degrees in computer science from The
University of Texas at Dallas, Richardson, USA. He is an Associate Professor with the
Department of Computer Science & Information Technologies, Frostburg State
University, Frostburg, MD, USA. His research interests include data security and
distributed systems. Dr. Xiao has published over 30 peer reviewed papers in prestigious
journals and peer reviewed conference proceedings.

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