Closed-form Queuing Models

• Closed-form queuing models are analytical methods that yield exact, recursive solutions for steady-state distributions, throughput, and waiting times under defined assumptions.
• They employ techniques such as product-form solutions, max-plus algebra, and recurrence relations to compute system dynamics without resorting to full simulation.
• These models enhance performance analysis by enabling efficient computation and optimization in network design, resource allocation, and capacity planning.

Closed-form queuing models provide analytical solutions for steady-state distributions, performance metrics, and system dynamics in queueing systems under specific structural assumptions. Such models enable exact or recursive calculation of quantities like queue-length distributions, throughput, waiting-time, and stability, circumventing the need for full simulation or purely numerical optimization. The variety and scope of closed-form queuing models are determined by the service discipline, arrival processes, network topology, buffer constraints, and any special operational mechanisms (e.g., skip-over policies, bulk service, fork-join synchronization).

1. Closed-Form Product-Form Solutions in Queuing Networks

Classic queueing networks like Jackson or BCMP networks admit product-form steady-state distributions under Markovian assumptions, typically Poisson arrivals and exponential service. More generally, in closed networks with finite capacity buffers and skip-over routing, the stationary distribution for a system state $n = (n_1,\ldots,n_M)$ (with %%%%1%%%% customers at station $i$ and $\sum n_i = N$) is given analytically via

$P(n)=\frac{1}{G} \prod_{i=1}^{M} f_i(n_i)$

where the normalizer $G$ is computed through a multi-dimensional convolution over all feasible customer allocations respecting buffer capacities $C_i$. The service functions $f_i(k)$ admit a closed-form recursive representation reflecting service demand and visit ratios. Recurrence for $G$ is stated as

$$g(n,m,\{c\}) = \sum_{k=0}^{\min(n,c_m)} f_m(k)\, g(n-k,m-1,\{c\})$$

with principled boundary conditions. Skip-over mechanisms and finite buffers adjust the domains, but the product-form persists, enabling marginal queue-length, utilization, and throughput calculations strictly algorithmically. Computational algorithms based on convolution and extended Mean Value Analysis offer complexity $O(MN C_{\max})$, supporting large-scale implementation (Balbo et al., 2024).

2. Server-Job Compatibility and Order-Independent (OI) Product Forms

In first-come-first-served (FCFS) systems with arbitrary job-server compatibility graphs, closed-form stationary distributions arise under Poisson arrivals, exponential service, and a bipartite compatibility relation. The core OI property states that the service rate assigned to the $j$th job is invariant under permutation and only depends on the historical set of classes. The resulting state distribution for job sequence $c_{\vec n}$ is

$$\pi^C(c_{\vec n}) = \pi^C(\emptyset) \prod_{i=1}^n \frac{\lambda_{c_i}}{\mu(c_{\vec i})}$$

where $\mu(c_{\vec i})$ is the (aggregate) service rate available to jobs up to $i$. Aggregate class-count marginal distributions are recursively computed using the function $\Phi(x)$, which itself admits closed-form recursions for important subclasses (e.g., nested systems, W-model). The closed-form sojourn time for class-$i$ is

$E[T^i] = \frac{E[\# \text{class-%%%%15%%%% jobs}]}{\lambda_i}$

and is further decomposable in compatible architectures (bridged, nested). Loss and admission control policies preserve the product-form and closed-form solutions (Gardner et al., 2020).

3. Max-Plus Algebraic State Equations for Fork-Join Networks

Fork-join queueing networks of arbitrary topology can be expressed as vector-valued max-plus linear state equations. For $n$ nodes and service times $\tau_{i,k}$ per node and job index,

$$d(k) = T_k \otimes G_0^T \otimes d(k) \oplus T_k \otimes d(k-1) \oplus T_k \otimes \bigoplus_{m=1}^{M_r} G_m^T \otimes d(k-m)$$

where $d(k)$ is the vector of departure epochs at step $k$, $T_k$ is the diagonal matrix of service times, and $G_m^T$ encodes immediate predecessor relationships and buffer statuses. Explicit finite-memory closed-form equations for $d(k)$ exist if and only if the “immediate-join” graph is acyclic. The resolvent $(E \oplus U)^p$ produces the closed form, with $p$ being the maximal path length in the graph. This algebraic framework accommodates blocking and buffer constraints by matrix extensions (via $H_m$), and readily computes exact departure times without simulation (Krivulin, 2012).

4. Bulk-Service Queuing Models with Markovian Disruptions

For bulk-service queueing with Poisson arrivals and service suspensions modeled as a continuous-time Markov chain, closed-form expressions for queue-length and waiting time metrics are derived. The headway between vehicles is distributed as a shifted difference of compound Poisson-exponential variables, with

$$E[H^{(n)}] = \bar{H}, \quad \text{Var}(H^{(n)}) = \frac{4\gamma T^{(n)}}{\theta^2}$$

and the number of arrivals during a headway is Poisson parameterized by this headway. The generating function of the queue length at boarding epochs, $Q(z)$, is given analytically,

$$Q(z) = \frac{\sum_{u=0}^C s_u \sum_{i=0}^u q_i (z^C - z^{C-u+i})}{z^C Y(z)^{-1} - \sum_{u=0}^C s_u z^{C-u}}$$

where $s_u$ represents the probability of $u$ boarding spaces and $Y(z)$ the arrival PMF. Mean and variance of queue length derive as derivatives of $Q(z)$ at $z=1$. The closed-form stability condition

$\rho^{(n)} = \frac{E[Y^{(n)}]}{E[S^{(n)}]} < 1$

dictates the regime for steady-state existence, and all formulas incorporate effects of disruptions analytically via the Markov generator and headway distribution (Mo et al., 2023).

5. Fluid and On–Off Source Models for TCP Buffers

Closed-form models for TCP traffic queues utilize a fluid “on–off” source abstraction, where each flow switches between “on” periods (transmitting a congestion window at wire-rate $\nu$) and “off” periods (waiting for RTT $T$). The aggregate input to a finite-buffer single-server queue is modeled as the superposition of $N$ such independent sources. Queue evolution follows

$\frac{dX(t)}{dt} = k\nu - C$

with $X(t)$ absorbed at $B$ (buffer size) and reflected at zero. The Markovian birth–death chain for active sources permits, in the exponential case, closed-form stationary and loss probability formulas, e.g.

$$p_{\text{loss}} = \pi_{\text{overflow}} = 1 - \sum_{k: d_k<0} \pi_k \left[1 - \exp(-|d_k| B/\sigma_k)\right]$$

where $d_k$ is the drift for $k$ active sources. For empirically observed truncated-normal on-durations (as in measured TCP window sizes), the stationary solutions are not in closed form and are computed numerically. Throughput per flow is then linked to loss probability via standard TCP laws. The model matches packet-level simulation quantitatively except under high flow synchronization, and exhibits limitations when non-exponential statistics, RED, or multi-bottleneck effects predominate (Genin et al., 2011).

6. Closed-Form Waiting-Time in Polling Systems with Renewal Arrivals

Polling systems, with one server cyclically visiting $N$ queues with renewal (non-Poisson) arrivals, admit closed-form approximations for mean waiting times spanning all traffic regimes using light-traffic and heavy-traffic asymptotics. Expansion at light traffic uses

$$E[W_{i}]^{LT} \approx E[S^{res}] + \rho_i (E[\hat{A}_i] \hat{g}_i(0) - 1) E[B_i^{res}] + \rho E[B^{res}] + (\rho-\rho_i)(E[S] - E[S^{res}]) + \frac{1}{E[S]} \sum_{k=i+1}^{i+N-1}\rho_k \sum_{j=i}^{k-1} V[S_j]$$

Heavy-traffic limit yields

$$E[W_i] = \frac{\omega_i}{1-\rho} + o((1-\rho)^{-1})$$

and a cubic interpolation yields a uniformly accurate closed-form for all $\rho$: $$E[W_{i,\mathrm{app}}](\rho) = \frac{K_{0,i} + K_{1,i}\rho + K_{2,i}\rho^2}{1-\rho}$$ with explicit expressions for $K_{0,i}, K_{1,i}, K_{2,i}$ in terms of the first and second moments of interarrival, service, and switch-over times. These formulas hold to within a few percent error across extensive numerical study for a general polling system, supporting capacity planning, optimization, and sensitivity analysis (Boon et al., 2014).

7. Methodological Rigour and Performance Implications

All closed-form models delineate specific application domains, stability conditions, and numerical regimes, with computational algorithms detailed for normalization constants, marginal distributions, throughput, and mean value analysis. Recurrence relations, matrix-algebraic transforms, and generating functions characterize the principal methodologies underpinning analytical tractability.

The primary assumptions—statistical homogeneity, Markovian input/service, acyclic topology, buffer constraints, and simplifications of operational rules—are motivated strictly by analytical solvability. Extensions to multiclass, priority, or non-Markov systems often require numerical or simulation-based approaches rather than closed-form. Numerical stability and computational complexity assessments are intrinsic to these methods, and remedies for underflow/overflow in MVA or convolution routines are precisely specified (Balbo et al., 2024).

Closed-form queuing models thus remain the cornerstone for both foundational analysis and applied system design in performance engineering, network optimization, transportation planning, and resource allocation, provided model assumptions are strictly controlled and validated against empirical regimes.