Abstract
A stochastic flow network composed of multistate arcs can be utilized to describe several practical systems such as computer networks, where transmission time taken for sending data to a sink is an important index. Determining a path with minimum transmission time is known as the quickest path problem (QPP). All algorithms addressing the QPP assume that the determined minimal paths (MPs) are disjoint. Further, for the general case of intersectional MPs, if a congestion phenomenon occurs during the transmission process, these algorithms will lead to an incorrect outcome. Moreover, in practical scenarios, as a budget limit is considered, spare routing is applied to consolidate the system. The objective is to develop an algorithm based on Monte Carlo simulations (MCSs) for evaluating the system reliability while considering the congestion phenomenon. The system reliability is the probability that a specific amount of data can be transmitted successfully through multiple MPs under both time and budget constraints. Furthermore, spare routing to increase the system reliability is established in advance to specify the main and spare MPs. Experiments validate the evaluation of system reliability based on MCSs. The credibility and efficiency of the proposed algorithm are also discussed.
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Abbreviations
 QPP:

Quickest path problem
 SFN:

Stochastic flow network
 MP:

Minimal path
 MCS:

Monte Carlo simulation
 ADF:

Allocation of data flow
 TANET:

Taiwan academic network
 n; n′ :

Number of arcs in SFN; number of arcs in the simplified SFN
 q; q′ :

Number of nodes in SFN; number of nodes in the simplified SFN
 a _{ i; } a _{ i } ′ :

Component i in SFN, i = 1, 2, …, n + q; component i in the simplified SFN, i = 1, 2, …, n′ + q′
 A; A′ :

{a_{i}1 ≤ i ≤ n}: the set of arcs in SFN; {a_{i}′1 ≤ i ≤ n′}: the set of arcs in the simplified SFN
 N; N′ :

{a_{i}n + 1 ≤ i ≤ n + q}: the set of nodes in SFN; {a_{i}′n′ + 1 ≤ i ≤ n′ + q′}: the set of nodes in the simplified SFN
 M _{ i; } M _{ i } ′ :

Maximal capacity of a_{i}, i = 1, 2,…, n; maximal capacity of a_{i}′, i = 1, 2, …, n′
 M; M′ :

{M_{i}1 ≤ i ≤ n}; {M_{i}′1 ≤ i ≤ n′}
 l _{ i; } l _{ i } ′ :

Lead time of a_{i}, i = 1, 2,…, n; lead time of a_{i}′, i = 1, 2, …, n′
 L; L′ :

{l_{i}1 ≤ i ≤ n}; {l_{i}′1 ≤ i ≤ n′}
 c _{ i; } c _{ i } ′ :

Transmission cost on a_{i}, i = 1, 2,…, n; lead time of a_{i}′, i = 1, 2, …, n′
 C; C′ :

{c_{i}1 ≤ i ≤ n}; {c_{i}′1 ≤ i ≤ n′}
 G; G′ :

(A, N, L, M, C): a SFN; (A′, N′, L′, M′, C′): a simplified SFN
 G :

Number of MPs in a group
 P _{ j } :

j^{th} MP in SFN, j = 1, 2, …, g
 V _{ e } :

e^{th} group of g MPs in SFN, e = 1, 2,…, u
 N _{ s } :

number of successes
 N _{ r } :

Number of runs
 d :

Total amount of data
 P _{ j } ′ :

j^{th} MP in the simplified SFN, j = 1, 2, …, g
 d _{ j } :

Amount of data assigned to P_{j}′, j = 1, 2, …, g
 s _{ i } :

Current capacity of a_{i}′, i = 1, 2, …, n′
 S :

(s_{1}, s_{2}, …, s_{n′}): a system state
 t :

Point in time
 T :

Time constraint
 B :

Budget limit
 d _{j}(t):

Amount of data assigned to P_{j}′ in data buffer at t, j = 1, 2, …, g, t = 1, 2, …, T
 m _{ j } :

Number of components on P_{j}′
 \( a_{k}^{j} \) :

k^{th} component on P_{j}′, j = 1, 2, …, g, k = 1, 2, …, m_{j}
 \( s_{k}^{j} \) :

Capacity of \( a_{k}^{j} \), j = 1, 2, …, g, k = 1, 2, …, m_{j}
 \( l_{k}^{j} \) :

Lead time of \( a_{k}^{j} \), j = 1, 2, …, g, k = 1, 2, …, m_{j}
 \( X_{k}^{j} (t) \) :

Amount of data assigned to P_{j}′ on \( a_{k}^{j} \) at t, t = 1, 2, …, T, j = 1, 2, …, g, k = 1, 3, …, m_{j} (odd number)
 \( Y_{k}^{j,c} (t) \) :

Amount of data assigned to P_{j}′ at c^{th} position inside \( a_{k}^{j} \) at t, t = 1, 2, …, T, j = 1, 2, …, g, k = 2, 4, …, m_{j} – 1 (even number), c = 1, 2, …, \( l_{k}^{j} \) – 1
 \( q_{k}^{j} \) :

Maximum available capacity of \( a_{k}^{j} \)
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Lin, YK., Huang, CF. & Chang, CC. Reliability of spare routing via intersectional minimal paths within budget and time constraints by simulation. Ann Oper Res (2021). https://doi.org/10.1007/s1047902003923y
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Keywords
 Spare routing
 Budget
 Monte Carlo simulation
 Reliability
 Intersectional minimal paths