In recent years, the cost of Internet access has fallen and the number of people using the Internet has increased. The number of people with Internet access and the Internet access speeds are different in different parts of the world [2, 3, 1].
Internet Architecture
On the Internet, access providers have limited bandwidth, and with this limited bandwidth they provide service to clients. In our model, we consider access providers with limited bandwidth that provide service to customers.
Multihomed clients
Customers connect to the Internet randomly and because of this randomness sometimes an ISP has a shortage of available bandwidth and sometimes an excess of available bandwidth. However, many times one internet service provider may be congested and another service provider may have sufficient bandwidth.
Motivation
Contribution of the thesis
A service provider must provide good service to customers and if it fails, it must pay fines to customers. When a customer requests a connection and the service provider denies the request, they pay a connection penalty.
Organization of the thesis
Static Pricing
- Flat Pricing
- Cumulus Pricing
- Usage Based Pricing
- Time of the day Pricing
- Priority Pricing Scheme
Since there are fewer clients in the higher class, the higher class clients get better QoS. When a user dials a number, if the network is not congested, his request will be placed in a conventional queue.
Dynamic Pricing
- Priority Pricing
- Auction Based Pricing
- Congestion Pricing
- Game Theory in pricing
When a client does not choose any service class, the service provider is fined. In [42], authors have considered a model where a service provider has limited bandwidth and clients make service requests.
Current Practice of ISPs
The paper talks about a game theory approach in which the users and the service provider try to maximize their income and reach equilibrium. The Afrikaans service provider MTN provides dynamic congestion pricing in which the cost of a call is determined every hour based on the level of usage.
Pricing with Service Level Agreements
- Symbol Declaration
- Nash Equilibrium solution method (Accurate Solution)
- Finding income per unit time at steady state
- Finding steady state probability
- Finding W aiting(i, s1)
- Accurate Solution Complexities
Our payoff function (or the criterion for deciding whether a solution is the best) is the expected income of the service provider in the steady state. Let I(1) be the steady-state income per unit of time of service provider 1 and let I(2) be the income per unit of time in the steady state of service provider 2.
Approximate solution
Steady State Probability
The combined steady-state probability P r(s1, s2) represents the probability that service provider 1 is in states1 and service provider 2 is in states2. In the approximate solution, the state of a service provider is represented by a single integer and therefore there are some differences in the steady-state probability formula.
Finding expected clients in session and waiting for session
- Finding E(i, m)
- Finding W aiting(i, s)
Approximate Solution Complexities
It is assumed that if a service provider charges price p from an arriving customer in state m, it will charge price p or less in state(x) givenx≤m. It is not expected that a service provider will charge a low price in state 5 and a high price in state 2.
Comparison of the two solutions
The method of choosing service provider 1's decisions and then finding the best decisions of service provider 2 and then finding the best decisions of service provider 1 and so on requires finding all possible values of C(). Therefore, to find all possible decision matrices, the goal is to find the values in which the decision changes from low prices to high prices, and when the highest price is reached, the decision becomes rejection.
Existence of Nash Equilibrium
As the rate of arrivals increases, the service provider is likely to increase the rates charged and/or deny connection requests from incoming customers. Similarly, if service provider 1 lowers prices or reduces the number of rejections, service provider 2 can also lower prices or reduce the number of rejections.
Limitations of Nash Equilibrium
N is a string representing the number of clients in the session, and its index ranges from 1 to e. R is a string representing the number of clients waiting for a session, and its index also ranges from 1 to e.
The Model
N An array representing the number of clients in a session and its index ranges from 1 to e. R An array representing the number of clients waiting for a session and its index also varies from 1 to e.
Accurate Solution
Solution Method
- Continuous Time Markov Decision Process
- AAEC (Advantage of An Extra Client) Method
- Method to find advantage of a decision in terms of D() 51
So the decision that provides the best benefit to an additional client is the decision whose value replaces the existing value of the current element. If C(state) and Cnew(state) are the same for every possible 'state', the optimal result is obtained.
Finding the value of D()
The first part concerns the function D1(), which represents the expected income a customer earns. The expected future income earned between time t and time t+dt depends on the client's condition at time t.
Expanded Equations of D1() and D2()
- Expanded Equations of D1
- Expanded equation of D2
The probability of an arrival that changes state from state 1 to state 2 in time dt)×D2t(state 2, t+dt). Probability of a client exiting the idle state that changes state from state1 to state2 at time dt).
Solution and Complexity analysis
- Solution and Complexity analysis
- Reduced complexities when e = 1
We find the complexity in terms of mmaks, e and B by finding the number of possible values of (m, N, R, K). This has reduced the number of possible conditions that need to be checked, but it is still quite large.
Proof that the AAEC method always produces an optimal so-
When e becomes 1, only one integer is used to store the number of clients in session and the number of clients waiting for session. As already mentioned, the state of a service provider is represented as (m, N, R) where m represents the number of customers connected.
Session Approximate Solution
- Finding solution C a ()
- Finding Income
- Finding W aiting()
- Finding P ri()
- Finding E d
- Complexity Analysis
Multiplying them all, the number of possible values of P r() formmax-connected clients is O(mmaxe×Be). Since m can vary from 0 tommax, this complexity must be multiplied by mmax to get the number of possible values of P r().
Grouped approximate solution
The number of instances of Ca() that will be generated will be of the order of mmaxT. The total time complexity is the maximum of the two, and it is O(mmaxe+1×Be+mmaxT+1).
A Simple Heuristic
Small ISP Simulation and variation of mean arrival rates
The quality of service is improved for both the exact solution and the approximate solution of the session. This shows that the approximate and exact solution of the session significantly improves the income and quality of the service.
Variation of bandwidth
The difference between the revenue of the exact solution and the approximate solution of the session is approximately within 10% of the revenue of the exact solution. This shows that the approximate session solution and the exact solution improve the revenue and quality of service on average.
Variation of Penalties
Simulation result shows that revenue produced by the accurate solution and the session approximate solution is about 50% more compared to the simple heuristic solution. The difference between revenue from the exact solution and the session approximate solution is approximately within 10 % of the revenue in the accurate solution.
Solutions for medium and large service providers
Variation of mean arrival rates in Medium ISP
Simulation results show that the estimated session solution improves income by 35% compared to the simple heuristic solution. The batched approximation solution produces virtually no delay, and the session approximation solution reduces the delay by about 36.
Variation of mean arrival rates in large ISPs
Variation of Bandwidth
Therefore, we also later perform simulations to determine which approximate solutions should be used and when. Simulations also show that when bandwidth is low, the performance difference between our solutions is large, and when bandwidth is high, the performance difference in our simulations is small.
Which approximate solution is better?
Effect of varying mean idle time on the two solutions
There are 9 simulations and in each simulation the average idle time value is different. As shown, when the average idle time is low, the pooled approximation performs better, and when it is high, the session approximation performs better.
Effect of congestion on the two solutions
Therefore, we expect that congestion will improve the performance of the grouped approximation solution compared to the session-approximate solution. When the bandwidth is low, the batched approximation solution performs better and when the bandwidth is high, the approximate session solution performs better.
Which solution should be used when?
This is done by estimating the average idle time, average session time and average waiting time. The results show that when the ratio of average idle time to average non-idle time is 6 or more, the performance of the session approximate solution is better and when it is less, the performance of the clustered approximate solution is better.
Evaluation of our simple heuristic solution
The number of customers arriving is
Realistic schemes
Scheme 1: Each client has access to two service providers
Based on the prices advertised by the service providers, the arriving customer chooses the service provider that offers the lower price. Thus, each service provider sees arrivals at a rate of 7 (5 × 1.4) customers per second, and for each arrival there is a single competing service provider.
Scheme 2: Each client has access to all service providers
For each of the simulations, the revenues, delays and rejections are added up and shown in the table. The simulation result shows that the grouped approach solution achieves maximum revenue among the three solutions and the improvement is 56% compared to the simple heuristic solution.
Scheme 3: A client consumed fixed bandwidth for complete
The approximate session solution can be executed for medium-sized service providers (when mmax'100 and e=2), but cannot be executed for large-sized service providers (whenmmax>1000ande≥2). Clustered Approximate Resolution and Session Clustered Approximate Resolution can work for large sized service providers.
Future Work
- Game Theory Solution
- Improvement of time and space complexities of our accurate
- More general model
- Inclusion of other QoS parameters
We also performed simulations for medium and large service providers and found that there was an improvement in revenue and quality of service in our exact and approximate solutions (approximate session solution, combined approximate solution and approximate session solution) as compared to simple heuristics in almost all simulations. One future direction is to present Nash equilibrium solutions for more than two service providers.
Publications
Mother of Invention: Network operators in the poor world are cutting costs and increasing access in innovative ways. A game-theoretic formulation of network selection in competitive wireless networks: An analytic hierarchy process model.
Internet autonomous systems
Multi-tier internet
Load balancing by Dynamic Pricing
Dynamic Pricing with Alternatives
The connection process
Connected clients
A service provider in our model
A service provider in our model
Access plans of an ISP
Symbol declaration
Comparison between Accurate and Approximate solutions
Definition of Symbols
Function declaration
Definition of Symbols
Simulation results show that the income of the exact solution and the session approximate solution are almost three times compared to the simple heuristic solution. The exact solution and the approximate session solution were compared with the simple heuristic solution for small and medium ISPs.
Definition of Symbols whose values are calculated
Simulation Details: Accurate-approximate-heuristic comparison 1
Simulation Result: Accurate-approximate-heuristic comparison 1
Simulation Details: Accurate-approximate-heuristic comparison 2
Simulation Result: Accurate-approximate-heuristic comparison 2
Simulation Details: Accurate-approximate-heuristic comparison 3
Simulation Result: Accurate-approximate-heuristic comparison 3
Simulation Details: Approximate-heuristic comparison 1
Simulation Result: Approximate-heuristic comparison 1
Simulation Result: Approximate-heuristic comparison 2
Simulation Details: Approximate-heuristic comparison 3
Simulation Result: Approximate-heuristic comparison 3
Simulation Details: Search for an appropriate approximate solution 1 119
Simulation Details: Evaluation of the simple heuristic
Simulation Result: Evaluation of the simple heuristic
Simulation Details: Realistic Scheme 1
Simulation Result: Realistic Scheme 1
Simulation Result: Realistic Scheme 2
Simulation Result: Realistic Scheme 3
Complexities of our Nash Equilibrium solutions
Complexities of our Non Game-theoretic solutions
