Problem Statement
A customer retention model between two competing cloud subscription tiers, Basic ($B$) and Premium ($P$), is represented by transition matrix $\mathbf{T}$:\n\n$$\mathbf{T} = \begin{pmatrix} 0.85 & 0.10 \\ 0.15 & 0.90 \end{pmatrix}$$
**(a)** [2 marks] Explain what the entry $0.15$ in the matrix represents in the context of customer migration.
**(b)** [3 marks] If initially $60\%$ of customers are on Basic and $40\%$ on Premium, find the market share distribution after 3 billing cycles.
**(c)** [3 marks] Find the long-term steady-state market share vector $\mathbf{s}$.
Verified Solution & Marking Scheme
Part (a) [R1]
The entry $0.15$ represents the probability that a Basic subscriber upgrades to Premium in the next cycle.
Part (b) [M1]
Compute $\mathbf{x}_3 = \mathbf{T}^3 \mathbf{x}_0 = \begin{pmatrix} 0.85 & 0.10 \\ 0.15 & 0.90 \end{pmatrix}^3 \begin{pmatrix} 0.60 \\ 0.40 \end{pmatrix}$
Part (b) [A1]
$\mathbf{T}^3 = \begin{pmatrix} 0.643 & 0.238 \\ 0.357 & 0.762 \end{pmatrix} \implies \mathbf{x}_3 = \begin{pmatrix} 0.481 \\ 0.519 \end{pmatrix}$
Part (b) [A1]
$48.1\%$ Basic, $51.9\%$ Premium
Part (c) [M1]
Steady state satisfies $\mathbf{T}\mathbf{s} = \mathbf{s} \implies \begin{pmatrix} 0.85 & 0.10 \\ 0.15 & 0.90 \end{pmatrix}\begin{pmatrix} s_1 \\ s_2 \end{pmatrix} = \begin{pmatrix} s_1 \\ s_2 \end{pmatrix}$ and $s_1 + s_2 = 1$
Part (c) [A1]
$0.85s_1 + 0.10s_2 = s_1 \implies 0.10s_2 = 0.15s_1 \implies s_2 = 1.5s_1$
Part (c) [A1]
$s_1 + 1.5s_1 = 1 \implies 2.5s_1 = 1 \implies s_1 = 0.40, \quad s_2 = 0.60$. Steady state is $40\%$ Basic, $60\%$ Premium.