This study is motivated by industries in which products can be partially prepared and stored before demand occurs, while demand is stock-dependent. We study a multi-server system in which the servers utilise their idle time to produce and store ‘preliminary services’ (PSs) in order to reduce customers’ sojourn time and, as a result, stimulating demand by creating the anticipation of a shorter sojourn time. In order to facilitate closed-form solutions, we analyse a Markovian queueing-inventory model and apply matrix geometric (MG) methods. In contrast to most applications in which the rate matrix R of the MG analysis is calculated numerically, our analysis enables derivation of explicit solutions for the entries of R and discovers their relation to Catalan numbers, allowing a rapid solution for large systems. Consequently, the system’s stability condition is readily obtained and shown to be identical to that of a regular M/M/s queue. Two models are developed: one for non-perishable PSs, the other for perishable ones. An economic analysis is provided for two case studies: a bike store and a pizza store. We observe that the reward has low sensitivity to deviation from the optimal PSs capacity and high sensitivity to deviation from the optimal promotional level.

Two main methods are used to solve continuous-time quasi birth-and-death processes: matrix geometric (MG) and probability generating functions (PGFs). MG requires a numerical solution (via successive substitutions) of a matrix quadratic equationA₀+RA₁+R²A₂= 0. PGFs involve a row vector $\vec{G}(z)$ of unknown generating functions satisfying $H(z)\vec{G}{(z)^\textrm{T}} = \vec{b}{(z)^\textrm{T}},$ where the row vector $\vec{b}(z)$ contains unknown “boundary” probabilities calculated as functions of roots of the matrixH(z). We show that: (a)H(z) and $\vec{b}(z)$ can be explicitly expressed in terms of the tripleA₀,A₁, andA₂; (b) when each matrix of the triple is lower (or upper) triangular, then (i)Rcan be explicitly expressed in terms of roots of $\det [H(z)]$ ; and (ii) the stability condition is readily extracted.