Managing the risk associated with bandwidth demand uncertainty Sverrir Olafsson Mobility Research Centre
Sverrir.Olafsson@bt.com Content • Uncertain bandwidth requirements – Quantification – Risk management • Implementation of capacity instalment process • Estimating time to capacity expiry • Optimal timing of capacity instalment • Use of real options
Sverrir.Olafsson@bt.com Bandwidth demand risk • “ Every day I look at the decision: should we build or should we lease ”? • Unknown demand for bandwidth – Uncertain future applications – Uncertain uptake of future applications – Uncertain customer base
Sverrir.Olafsson@bt.com Bandwidth price risk • Unknown price of bandwidth – Prices are on their way down – The rate of decline is unknown ⇒ risk
Sverrir.Olafsson@bt.com What uncertainties? • Possible scenarios Demand known Analogy to commodity market were Uncertainties magnitude is known but the price • Price of bandwidth of commodity is unknown Uncertainties Analogy to commodity market were • Required bandwidth both the magnitude and price of • Price of bandwidth commodity are unknown • Require different risk management • Most risk management procedures assume – Known demand (currency, commodity,…) – Uncertain price
Sverrir.Olafsson@bt.com Aim of risk management • Identify risk sources • Quantify the risk caused • Control the risk – Hedge • only some risks can be hedged (commodities markets,….) – More efficient decision making • operational caution • insurance • real options
Sverrir.Olafsson@bt.com Bandwidth demand risk • Required bandwidth can be acquired in different ways – Building networks – Install additional capacity, lit fibre – Lease – Enter derivative contracts (futures, swaps, options) • Before deciding on action – Model bandwidth demand evolution – Model price evolution
Sverrir.Olafsson@bt.com Modelling bandwidth demand • Evidence for “exponential growth” • But, rate of growth is uncertain • Model as geometric Brownian motion = η dW dt = µ + σ dD D dt D dW t t ( ) t t t t η ∈ N 0 , 1 • Therefore t x 0 = 50, µ = 0.3, σ = 0.5 140 120 120 Process Mean 1 Standard deviation = µ − σ + σ 120 100 100 2 D D exp t W t 0 t Standard deviation 100 8 0 8 0 2 Value 80 6 0 6 0 [ ] [ ] = ( ) = µ E D D exp t ; var D 60 4 0 4 0 t 0 t 40 2 0 2 0 20 0 0 0 0 0 100 100 100 200 200 200 300 300 300 400 400 400 500 500 500 600 600 600 700 700 700 800 800 800 900 900 900 1000 1000 1000 Time [days] gbmo(50,0.3,0.50,1/360,1000,1)
Sverrir.Olafsson@bt.com Possible demand scenarios • Monte Carlo simulations – iterate a large number of scenarios each compatible with the assumptions Capacity evolution, µ = 0.3, σ = 0.25, D 0 = 50 250 200 Required capacity 150 100 50 justgbm.m 0 0 100 200 300 400 500 600 700 800 900 1000 Time (Days)
Sverrir.Olafsson@bt.com Assumptions and questions • Given – Present demand D 0 – Presently available capacity C 0 , D 0 < C 0 • Questions – What is the probability of exceeding the installed capacity within a given time? – What is the proper capacity instalment rate? • Contrast the expected life of presently installed capacity with expectations about price evolution
Sverrir.Olafsson@bt.com Probability of exceeding capacity • Probability of exceeding installed capacity • Probability density function • Cumulative probability function Parameter estimates, a = 6.2112, b = 141.9538 µ = 0.35, σ = 0.25, Initial demand = 50, Capacity = 150 1 Empirical data 0.035 Log-normal Gamma fit Empirical Gamma 0.03 0.8 Probability density Cumulative probability 0.025 0.6 0.02 0.015 0.4 0.01 0.2 0.005 0 0 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 3000 gmbreach.m gmbreach.m Days Days
All scenarios have the same mean capacity life Sverrir.Olafsson@bt.com 1 C Impact of uncertainty = 0 log t µ D 0 µ = 0.5 , σ = 0.2 µ = 0.5 , σ = 0.4 µ = 0.5 , σ = 0.8 µ = 0.5, σ = 0.8, Initial demand = 50, Capacity = 150 µ = 0.5, σ = 0.2, Initial demand = 50, Capacity = 150 µ = 0.5, σ = 0.4, Initial demand = 50, Capacity = 150 0.09 0.05 0.05 0.045 0.08 0.045 0.04 0.07 0.04 0.035 Probability density Probability density Probability density 0.06 0.035 0.03 0.05 0.03 0.025 0.025 0.04 0.02 0.02 0.03 0.015 0.015 0.02 0.01 0.01 0.01 0.005 0.005 0 0 0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Days Days Days µ = 0.5, σ = 0.8, Initial demand = 50, Capacity = 150 µ = 0.5, σ = 0.2, Initial demand = 50, Capacity = 150 µ = 0.5, σ = 0.4, Initial demand = 50, Capacity = 150 1 1 1 0.9 0.9 0.9 0.8 0.8 0.8 Cumulative probability Cumulative probability Cumulative probability 0.7 0.7 0.7 0.6 0.6 0.6 0.5 0.5 0.5 0.4 0.4 0.4 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0 0 0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Days Days Days
Sverrir.Olafsson@bt.com Justification for log-normal modelling • The empirical cumulative probability is well ) ( ) ( − µ − σ 2 1 log C / D / 2 t ( ) approximated by the ≤ = + 0 CP D C , t 1 erf σ t log-normal distribution 2 2 t µ = 0.25, σ = 0.4, Initial demand = 50, Capacity = 150 1 • Deviations are explained by demand 0.8 exceeding installed capacity and then go Cumulative probability down below installed capacity again 0.6 0.4 0.2 Log-normal Empirical Gamma 0 0 1000 2000 3000 4000 5000 6000 Days
Sverrir.Olafsson@bt.com Criteria for delaying instalment • There are benefits in delaying the acquirement of additional capacity – Cost – Efficient usage • Risks – QoS reduction – Loss of customers • Quantifying the criteria requires assumptions on the price evolution
Sverrir.Olafsson@bt.com Bandwidth demand risk • Bandwidth evolution is a stochastic process D ( t ) • Match installed capacity optimally to demand • Upgrade sequence { } Ω = ∆ ∆ , ,..., , ;... t C t C 1 1 k k • The process C ( t ) should stochastically dominate D ( t ) ( ( ) ( ) ) ≥ → Pr C t D t 1 • Approach – Dynamic programming, simulation, real options
Sverrir.Olafsson@bt.com Controlled instalments • Probability to exceed installed capacity ) ( ) ( − µ − σ 2 1 log C / D / 2 t ( ) ≤ = + 0 0 CP D C , t 1 erf σ t 0 2 2 t → = + D D D dD [ GBM , uncontroll ed ] + t t 1 t t → = + C C C dC [ controlled instalment ] + T T 1 T T • The instalment process will depend on – Expected growth – Expected volatility – Required QoS
Sverrir.Olafsson@bt.com Instalment strategy • Probability that demand does not exceed installed capacity for different instalment strategies Probability that demand is below the installed capacity Initial demand = 50, Initial capacity = 100, µ = 0.2, σ = 0.50 Probability that demand is below the installed capacity Initial demand = 50, Initial capacity = 100, µ = 0.2, σ = 0.50 1 1 0.95 0.9 0.9 0.8 0.85 µ inc =0.05 µ inc =0.05 0.8 µ inc =0.10 µ inc =0.10 0.7 µ inc =0.15 µ inc =0.15 0.75 µ inc =0.20 µ inc =0.20 0.6 µ inc =0.25 µ inc =0.25 0.7 µ inc =0.30 µ inc =0.30 0.65 0.5 0 200 400 600 800 1000 0 1000 2000 3000 4000 5000 Program:tempcapincrease.m Days Days
Sverrir.Olafsson@bt.com Time to capacity exhaustion • Assume additional capacity is installed at the average rate µ i σ 2 ( ) ( ) = µ − + σ = µ D D exp t W t t C exp t t 0 0 in 2 • Time to exhaustion C C 0 0 log log D D ( ) = ⇒ = 0 0 t E t σ σ ( ) 2 2 = ( ) E W 0 µ − µ − + σ µ − µ − W t i i 2 2
Sverrir.Olafsson@bt.com Different instalment strategies • Installing capacity at different – Rates – Time intervals Initial demand =50, Initial capacity = 75, µ = 0.35, σ = 0.25 Initial demand =50, Initial capacity = 75, µ = 0.35, σ = 0.25 100 100 Capacity coverage [%] Capacity coverage [%] 98 95 96 90 94 85 92 80 90 75 200 600 150 50 50 400 40 40 100 30 30 20 200 50 20 10 10 0 0 Days between instalments 0 0 Excess instalment [%] Days between instalments Excess instalment [%] capacityplot([0:0.05:0.45],[1:50:200],50,75,0.35,0.25,1/360,1000,1000,1.5); capacityplot([0:0.05:0.45],[1:50:500],50,75,0.35,0.25,1/360,1000,1000,1.5)
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