Computing transient probabilities in Markovian queues with balking conditional on a fixed number of joined customers

Abstract: We analyze the transient behavior of a Markovian queue with balking, conditional on exactly $K$ customers joining the system in a finite time interval $[0, T]$. A key quantity of interest is the cumulative number of balking customers, for which we consider the probability generating function (PGF) and derive an equation it satisfies. This approach circumvents the computational burden arising from dealing with high-dimensional Markovian system, which arises when attempting to directly compute the joint distribution of the cumulative number of balking customers together with the total number of joining customers and the number of customers in the system. To compute state transitions in $(t, T]$ efficiently, we examine the conditional state probability at time $T$ given the system state at time $T - u$, and show that it satisfies a linear differential equation in $u$. For piecewise-constant arrival rates, we develop a numerical procedure for evaluating the moment of the total number of balking customers, jointly with the cumulative number of joining customers and the number of customers in the system, under the condition that $K$ customers joined. We also present numerical examples that highlight counterintuitive behaviors arising from this conditioning, along with explanations of the underlying mechanisms.

• The paper introduces a novel differential equation method to characterize transient behavior in Markovian queues with balking, conditioned on a fixed number (example: 100 customers) of joined customers by time T.
• The method, involving a probability generating function-based differential equation and a backward-in-time ODE, transforms potentially time-long computational expressions into finite systems to estimate the conditional probabilities efficiently.
• Numerically, the study shows that this method results in clearer insights for the system dynamics about transient behavior for queue length, in contrast to steady-state analyses; for instance, arrival- and departure-rate dynamics can display unexpectedly unimodal behavior even under constant arrival rates.

Model and problem setting

The paper studies a Markovian queueing system in which customers arrive according to a nonhomogeneous Poisson process over a finite interval $[0,T]$, and an arriving customer who observes $\ell$ customers in the system joins with probability $\beta_\ell$ and balks otherwise. The service mechanism is Markovian, represented by a finite phase set $\mathcal{S}$, so the process $(A^{\text{join}}(t), L(t), S(t))$ is a continuous-time Markov chain (CTMC) on a state space of size $O(K^2|\mathcal{S}|)$ when restricted to at most $K$ joins. The object of interest is the transient joint distribution

$$\pi_{k,\ell,s,b}(t\mid K) = \Pr[A^{\text{join}}(t)=k,\ L(t)=\ell,\ S(t)=s,\ A^{\text{balk}}(t)=b \mid A^{\text{join}}(T)=K],$$

that is, system behavior conditional on exactly $K$ joined customers by time $T$. The authors motivate this conditioning as a complement to conventional time-averaged or transient analyses: $\ell$0 is an observable summary statistic (e.g., over one business day) even though it is not controllable. This work extends prior research on queues fed by exactly $\ell$1 i.i.d.-timed arrivals [Minh1977; Louchard1988; Louchard1994; Honnappa2015; Bet2019; Bet2020; Mandjes2025], which did not incorporate balking.

A naive approach requires an expanded state space including the cumulative balk count, which is countably infinite, or treating balking as a Markovian counting process driven by the underlying chain [Lucantoni1985; Takine1994]; either way, the state space grows as $\ell$2 — already exceeding 10,000 states for modest values such as $\ell$3, $\ell$4. A second difficulty is that computing the conditioning factor $\ell$5 repeatedly across many time points is expensive.

Generating-function characterization

The paper circumvents the first difficulty by working with the probability generating function (PGF) of the cumulative balk count rather than its full distribution. The central structural result (Lemma 1) is a first-order differential equation for the unconditional PGF vector:

$\ell$6

where $\ell$7 is a diagonal matrix collecting the balking probabilities $\ell$8. Differentiating $\ell$9 times at $\beta_\ell$0 yields a closed recursion for factorial moments $\beta_\ell$1 driven by the lower-order moment through $\beta_\ell$2 (Lemma 3). This reduces the infinite-dimensional counting-process computation to a family of finite ODEs of the same size as the base CTMC.

The second difficulty is resolved by a backward-in-time formulation. Defining $\beta_\ell$3, Lemma 2 shows this quantity satisfies the linear ODE $\beta_\ell$4 with boundary condition concentrated on states with $\beta_\ell$5 at $\beta_\ell$6. Solving once backward over $\beta_\ell$7 then provides the conditioning factor for all intermediate times simultaneously. Theorem 1 combines these pieces into

$\beta_\ell$8

and Theorem 2 gives the analogous identity for factorial moments $\beta_\ell$9. This factorization exploits the fact that $\mathcal{S}$0 and $\mathcal{S}$1 are conditionally independent given $\mathcal{S}$2, so no joint tracking of balks across the conditioning event is required.

Computational procedure for piecewise-constant rates

For piecewise-constant $\mathcal{S}$3 and $\mathcal{S}$4, all quantities reduce to matrix exponentials, which are evaluated by uniformization to ensure numerical stability (all operations involve only non-negative additions and multiplications, avoiding cancellation error). A notable technical point concerns the stacked matrix

$\mathcal{S}$5

used to advance the state probability and the first $\mathcal{S}$6 factorial moments jointly: because its row sums need not vanish, it is not a proper generator. The authors handle this by factoring out $\mathcal{S}$7 and choosing the uniformization rate as $\mathcal{S}$8, which guarantees that both $\mathcal{S}$9 and $(A^{\text{join}}(t), L(t), S(t))$0 are substochastic — a necessary condition for uniformization to be valid. Exploiting the block structure of $(A^{\text{join}}(t), L(t), S(t))$1, each power application decomposes into per-block recurrences, and sparsity of the M/PH/1 generator (detailed in an appendix) further reduces per-iteration cost. The overall algorithm computes $(A^{\text{join}}(t), L(t), S(t))$2 forward and $(A^{\text{join}}(t), L(t), S(t))$3 backward, then assembles conditional moments via the product formula; truncation depth $(A^{\text{join}}(t), L(t), S(t))$4 of the Poisson series is an explicit accuracy parameter.

Numerical findings

The examples concern an M/PH/1 queue with mean service time one, $(A^{\text{join}}(t), L(t), S(t))$5, $(A^{\text{join}}(t), L(t), S(t))$6, and entry probabilities $(A^{\text{join}}(t), L(t), S(t))$7. The headline observation is that conditioning on $(A^{\text{join}}(t), L(t), S(t))$8 induces persistent transient dynamics even under a time-homogeneous arrival rate: for $(A^{\text{join}}(t), L(t), S(t))$9, the expected number in system rises and then falls (a unimodal trajectory), whereas for smaller $O(K^2|\mathcal{S}|)$0 it remains nearly flat. This contradicts the intuition inherited from unconditioned models, where the system converges toward steady state.

The authors trace the mechanism carefully. Decomposing $O(K^2|\mathcal{S}|)$1, they show the conditional joining rate exhibits a systematic bias — when $O(K^2|\mathcal{S}|)$2, the constraint forces many balking events, so the conditioned path admits more customers early, raising congestion and creating balking opportunities. Crucially, they attribute this bias primarily to the arrival process itself ($O(K^2|\mathcal{S}|)$3 is time-varying) rather than to balking, whose conditional rate simply tracks congestion. A diagnostic emerges from comparing $O(K^2|\mathcal{S}|)$4 with $O(K^2|\mathcal{S}|)$5: when the unconditional expected join count exceeds $O(K^2|\mathcal{S}|)$6, trajectories peak mid-interval; when it falls short, they increase steadily.

Service-time variability shifts where the bias resides. At fixed $O(K^2|\mathcal{S}|)$7 (so $O(K^2|\mathcal{S}|)$8): large coefficients of variation produce frequent long services, heavy congestion, and elevated departure-rate bias, while small $O(K^2|\mathcal{S}|)$9 shifts the bias to the arrival process, with the departure rate saturating near capacity. The shape of $K$0 matters as well: concave entry probabilities concentrate entries early with late-peaking balking, while convex $K$1 produces milder congestion peaks, fewer arrivals, more departures, and fewer balks, since the system responds gradually rather than by inflating congestion to induce balking. All five tested $K$2 forms were calibrated to match unconditional expected balk counts in $K$3, isolating shape effects from aggregate balking intensity.

Limitations and open questions

The analysis assumes that once a customer joins, she remains until service completion; reneging is excluded. Consequently, conditioning on the number of served customers does not coincide with conditioning on the number of entries under abandonment, and the framework does not directly apply to systems with early departures — the authors identify this extension explicitly as future work. The numerical procedure is developed only for piecewise-constant arrival rates and generators; general time-varying parameters would require alternative ODE solvers whose stability properties are not addressed here. Finally, the truncation parameter $K$4 of the uniformization series is treated as an input without formal error bounds tied to it, and the reported experiments are limited to single-server M/PH/1 queues with specific $K$5 calibrations.

Conclusion

This paper provides an exact, computationally tractable characterization of transient behavior in Markovian balking queues conditioned on a fixed number of joined customers. The two key devices — a PGF-based differential equation that eliminates explicit tracking of the balk counter, and a backward-in-time ODE for the conditioning probability — together reduce a high-dimensional conditional computation to forward and backward solves of finite linear systems, stabilized via uniformization. The numerical study demonstrates that this conditioning systematically distorts arrival and departure dynamics, producing unimodal queue-length behavior even under constant arrival rates, and identifies whether the distortion appears in arrivals or departures as a function of service variability and entry-probability shape.