M/M/c Queue Model: Analysis & Extensions

• M/M/c Queue Model is a continuous-time, Markovian multi-server queue characterized by Poisson arrivals, exponential service times, and Erlang’s C formula.
• The model enables closed-form computation of key metrics such as server utilization, waiting probability, and mean sojourn time, supporting robust system design.
• Extensions include priority schemes, state-dependent service, and extreme-value analysis, broadening its application in telecommunications and operational research.

The $M/M/c$ queue is a classical continuous-time, Markovian multi-server queueing model central to queueing theory, performance analysis, and operations research. Defined by Poisson arrivals, $c$ parallel servers, and independent exponential service times, it underlies a wide range of performance models in e.g. telecommunications, call centers, and service systems. Recent research continues to extend, compute, and apply its probabilistic properties: from joint queue-length distributions and extreme-value statistics to busy-period durations, multi-priority systems, and state-dependent service mechanisms.

1. Formal Model Description

Let arrivals form a Poisson process of rate $\lambda$. There are $c$ identical servers, each independently serving one customer at exponential rate $\mu$. The system allows infinite waiting room; arrivals that find fewer than $c$ customers in service immediately begin service, while those finding all $c$ servers occupied join a first-come-first-served backlog queue. Stability requires $\rho := \lambda/(c\mu) < 1$. The system state is $X(t) = n$ ($0 \le n < \infty$), the total number of customers present. The process is birth–death, with rates: $\lambda_n = \lambda \quad \forall n,$

$$\mu_n = \begin{cases} n\mu, & n \le c \ c\mu, & n \ge c \end{cases}$$

Balance equations yield the steady-state probabilities $P_n = \lim_{t\to\infty}\mathbb{P}\{X(t)=n\}$. The normalization is: $$\frac1{P_0} = \sum_{j=0}^{c-1} \frac{(c\rho)^j}{j!} + \frac{(c\rho)^c}{c!(1-\rho)}$$ The explicit forms are: $$P_n = \begin{cases} P_0 \frac{(c\rho)^n}{n!}, & n < c \ P_0 \frac{(c\rho)^c}{c!}\rho^{n-c}, & n \geq c \end{cases}$$ The probability that an arrival must wait (all servers busy) is: $P\{\text{wait}\} = \frac{(c\rho)^c}{c!(1-\rho)}P_0$ This is Erlang’s $C$ formula and is foundational across contexts requiring multi-server delay analysis.

2. Key Performance Metrics and First-Order Properties

The principal performance metrics and their closed forms are as follows:

Measure	Expression	Comments
Utilization per server	$\rho = \frac{\lambda}{c\mu}$	$< 1$ for stability
Mean number in queue ($L_q$)	$$L_q = P\{\text{wait}\}\frac{\rho}{1-\rho} = \frac{(c\rho)^c}{c!(1-\rho)^2}P_0$$	Excludes those in service
Mean queueing time ($W_q$)	$W_q = L_q/\lambda$	Little’s law
Mean system size ($L$)	$L = L_q + \lambda/\mu$	$L_q$ plus those in service
Mean sojourn time ($W$)	$W = W_q + 1/\mu$	Includes service time
Probability all servers idle ($P_0$)	$[\,\text{see above}\,]$	Explicit sum

These explicit steady-state forms enable closed-form, numerically stable computation for arbitrary $(\lambda, \mu, c)$, rendering the $M/M/c$ queue tractable for analytic, design, and simulation contexts (Finch, 2019).

3. Distributional and Extreme-Value Properties

Beyond stationary means, recent work has addressed the high-level behavior of the queue, particularly the distribution of the maximum queue length over a finite interval. Let $$M_n = \max_{t\in [0, n]}\{\text{queue length at } t\}$$ (excluding those in service). The Poisson clumping heuristic provides an accurate asymptotic approximation for this extreme-value distribution.

Let $q = \lambda/(c\mu)$ and define $k = \log_{1/q}(n) + h + 1$. The probability that the maximum does not exceed $k$ is, for large $n$,

$$\Pr\{M_n \leq \log_{1/q}(n) + h\} \approx \exp\left[-C\, q^{h+1}\right]$$

where

$$C = c^{c-2}\lambda\mu^{c-3}(c\mu - \lambda)^2 \sum_{j=1}^c j! \binom{c-1}{j-1} \lambda^{c-j}\mu^{j-1}$$

Higher-order (finite-$n$) corrections introduce an additional bracketed term inside the exponential, improving the fit for moderate $n$ ($10^3$–$10^4$). The limit law is a discrete Gumbel and governs extremes for general $c$ (Finch, 2019). Mean and variance estimates are given by: $$\mathbb{E}[M_n] \approx \frac{\ln n}{\ln(1/q)} + \frac{\gamma + \ln(C q)}{\ln(1/q)} + \frac{1}{2},$$

$$\mathrm{Var}(M_n) \approx \frac{\pi^2/6}{[\ln(1/q)]^2} + \frac{1}{12}$$

These expressions match simulation results to a high degree of accuracy.

4. Extensions: Priority Classes and State-Dependent Service

Priority Extensions

Multi-class and multi-priority $M/M/c$ models have been extended to both non-preemptive and preemptive regimes.

Non-preemptive priority with $K$ classes: Poisson arrivals at rates $\lambda_\kappa$, exponential services, $c$ servers. Zuk and Kirszenblat (Zuk et al., 2023) derive explicit forms for the joint queue-length generating function $G_0(z_1,\ldots,z_{K-1})$ as a product of root-ratio terms, and invert it via multidimensional FFT to recover the joint stationary distribution. The fixed-point iteration for the vectorized stationary balance equations is robust and probability-preserving, with convergence slowed only as $r\to 1$ or $K$ increases. The methodology yields all joint and marginal performance measures and has been numerically validated to high precision.

Preemptive two-class priority: The continuous-time Markov structure yields Laplace transforms for all finite-time transition probabilities via spectral methods, including explicit closed forms for the vertical boundary (states with $j\ge c$ high-priority customers). The general case is recursively tractable with Ramaswami’s formula adapted from $M/G/1$ Markov chains (Selen et al., 2016).

State-Dependent and Time-Dependent Service

Service rates affected by experienced waiting time are considered in (D'Auria et al., 2021): each arrival is "tagged" by whether its virtual waiting time at entry exceeds a threshold $\tau$, and is served at either rate $\mu_1$ (if $W\leq\tau$) or $\mu_2$ (if $W>\tau$). The model is formulated via a Markov process tracking the virtual queueing time and server occupancy, matched to an embedded process with piecewise ODE structure. The stationary distribution is characterized recursively and in closed form as a mixture of exponentials for the waiting time, with rates determined as zeros of characteristic polynomials tied to the underlying Markov dynamics. When $\mu_1\ne \mu_2$, performance can change drastically, with slowdown ($\mu_2<\mu_1$) causing heavy-tailed delays.

5. Partial Busy Periods and Time-Dependent Behavior

The partial busy period ($T_{\mathrm{bp}}$): the time from system entry by an arrival (when queue was empty) until the system empties again, is analyzed explicitly for general $c$ in (Zuk et al., 2023). For $c=2$ the result connects to classic Karlin–McGregor integral representations. For arbitrary $c$, two main techniques are provided:

• Spectral method: The Laplace transform $\phi_1(s)$ is written in terms of resolvent sums over eigenvalues of the finite-dimensional generator, leading to a survival function as a sum of exponentials (from simple poles) plus a branch cut integral.
• Algebraic method: The closed-form representation involves a sum over Marcum Q-functions, with parameters and coefficients arising from roots of a uniquely defined "cut polynomial" of degree $2c-3$.

Large-$c$ asymptotics distinguish two regimes: fixed-traffic intensity ($r<1$) gives a mixture of empty and exponentially-distributed periods; the $M/M/\infty$ scaling yields mixtures tied to zeros of Kummer functions.

Efficient computation leverages eigenvalue routines (e.g., LAPACK’s {\tt dstevd}), Chebyshev quadrature, and FFT-based polynomial root solvers. These approaches allow full distributional computation for $c$ up to $O(10^2)$.

6. Analytical and Computational Methods

Explicit forms for steady-state and transient distributions are available in the classical case; closed-form, efficiently computable representations for time-dependent, multi-class, and state-dependent extensions require:

• Reduction to birth–death or block-tridiagonal Markov chains;
• Laplace transform (spectral) analysis, inversion via residue and cut contours;
• Explicit, finite-sum representations (Marcum-Q, Bessel series, or mixture of exponentials) using parameter polynomials derived from model structure;
• Fixed-point or matrix-analytic recursions (Ramaswami-like algorithms) for balance equations in high-dimensional spaces;
• Accelerated numerical inversion (FFT-based or Gauss–Chebyshev quadrature) for generating functions and transforms, maintaining probability normalization and exploiting structure for tractability;
• Direct simulation for empirical validation, as in the extreme-value paper of (Finch, 2019).

Accuracy, robustness, and computational efficiency are achieved up to $c \sim 10^2$ and, for priority models, $K\sim 3$–$5$.

7. Applications and Implications

The $M/M/c$ model and its extensions directly address:

• Performance guarantees in service systems (delay probabilities, maximal queue length, mean waiting times);
• Regenerative simulation interval estimation and confidence procedures (busy period distributions) (Zuk et al., 2023);
• Service system design; e.g., setting $c$ to meet SLA thresholds (via Erlang $C$ or joint queue-length measures);
• Priority queueing in telecommunications, health care, or call centers (preemptive vs non-preemptive delay, fairness, overflow probabilities);
• Queueing-time dependent dynamical control policies and their impact.

The collective body of work reaffirms the tractability and flexibility of the $M/M/c$ queue for both theoretical and applied queueing analysis, while delineating the frontiers of what is computable: joint queue-lengths for many classes require FFT inversion; busy periods demand spectral tools; maxima require clumping heuristics. Generalizations for non-Markovian service and arrival, reneging, or blocking remain challenging.

8. Context and Future Research

Recent advances yield the most explicit known formulas for joint queue-lengths (Zuk et al., 2023), partial busy periods (Zuk et al., 2023), and time-dependent multi-class probabilities (Selen et al., 2016) for $M/M/c$ systems. These results fill longstanding gaps, especially for $c>2$ and general $K$.

A plausible implication is that similar algebraic–spectral hybrid techniques could be adapted to $M/G/c$ or $GI/M/c$ queues, though the loss of Markov structure presents substantial obstacles. As system parameters scale (large $c$, near-critical $\rho\to 1$), asymptotic analysis and numerically stable algorithms remain an ongoing research frontier.

The $M/M/c$ queue retains a central role in queueing theory both as a reference model and as a basis for extendable, analytically tractable specializations in modern stochastic network analysis.