M/M/1 Feedback Queue Analysis

• M/M/1 Feedback Queue is a single-server model with Poisson arrivals and exponential service where Bernoulli feedback recycles customers, altering the effective service rate.
• The model bridges classical queueing theory with timing-channel communication, leveraging exponential memorylessness to analyze both congestion dynamics and information capacity.
• Research in this area explores strategic joining, threshold equilibria, and welfare paradoxes, providing insights into the trade-offs between individual incentives and overall system performance.

Searching arXiv for relevant papers on M/M/1 feedback queues and queue timing channels. An $M/M/1$ feedback queue is a single-server exponential-service queue in which “feedback” denotes a return of information or customers to an earlier stage of the system. In the queueing literature, the term most commonly refers to an observable FCFS system with Poisson arrivals, exponential service times, and instantaneous Bernoulli feedback: after a service completion, a customer departs with probability $q$ and otherwise immediately rejoins the end of the queue. In the queue-channel literature, the same phrase can refer to a continuous-time single-server timing channel with exponential service, where the transmitter encodes information into packet arrival times and causally observes past departures. These two usages share the same exponential single-server core, but they study different objects: congestion and equilibrium behavior in the former, and information capacity in the latter (Fackrell et al., 2021).

1. Terminological scope and canonical formulations

Two distinct models are central to the modern arXiv treatment of the $M/M/1$ feedback queue.

The first is the observable $M/M/1$ feedback queue with Bernoulli feedback. Its primitive parameters are a Poisson arrival rate $\lambda$, exponential service rate $\mu$, and a success probability $q\in(0,1]$. Customers are served in FCFS order. After each service completion, the customer departs successfully with probability $q$, and with probability $1-q$ the service fails and the customer immediately rejoins the end of the queue. This is the model used for strategic joining, balking, and reneging analyses (Fackrell et al., 2021).

The second is the timing communication through a single-server queue specialization relevant to an $M/M/1$ feedback query. In that setting, packet $q$0 arrives at time $q$1, departs at time $q$2, and

$q$3

where $q$4 is the waiting time and $q$5 is the service time. The service times are i.i.d. with mean $q$6, and in the continuous-time $q$7 specialization,

$q$8

Here feedback means that the transmitter causally observes past output timing, rather than customers physically returning to the queue (Aptel et al., 2017).

A common misconception is that the label $q$9 forces Poisson arrivals in every formulation. In the strategic queueing model, arrivals are indeed Poisson. In the timing-channel model, however, the encoder generally chooses arrivals strategically to communicate, and the paper explicitly notes that Poisson arrivals are not essential in the coding formulation (Aptel et al., 2017).

2. Queueing dynamics of Bernoulli feedback

In the Bernoulli-feedback model, the state variable is $M/M/1$0, the number of customers in the system at time $M/M/1$1. For the $M/M/1$2-th arrival at time $M/M/1$3, the joining position is

$M/M/1$4

with $M/M/1$5 meaning immediate service. Because each service attempt succeeds with probability $M/M/1$6, the number of service attempts required by a customer is geometric with mean

$M/M/1$7

Accordingly, the mean total service requirement per customer is effectively

$M/M/1$8

so the queue behaves, at the traffic level, as if the effective service rate were $M/M/1$9 rather than $M/M/1$0 (Fackrell et al., 2021).

This interpretation appears directly in the stability condition. When all customers always join and reneging is not allowed, stability holds iff

$M/M/1$1

Under that condition, the stationary queue-length distribution is geometric: $M/M/1$2 with effective load

$M/M/1$3

Thus Bernoulli feedback increases congestion by reducing the effective service completion rate from $M/M/1$4 to $M/M/1$5 (Fackrell et al., 2021).

For the always-join benchmark, the conditional expected sojourn time of an arrival who finds herself joining in position $M/M/1$6 is

$M/M/1$7

The paper identifies this as the key explicit queueing-performance formula. It is linear in $M/M/1$8, increases as $M/M/1$9 falls, and reflects the net service margin under feedback (Fackrell et al., 2021).

The 2025 extension retains the same underlying queue but shifts attention from mean delay to the whole sojourn-time distribution. The tagged-customer state space becomes

$\lambda$0

where $\lambda$1 is absorbing and denotes successful departure. This formulation is designed for payoff criteria that depend on $\lambda$2 and, through Laplace inversion, on the full cdf of $\lambda$3 (Taylor et al., 15 Jul 2025).

3. Timing-channel interpretation and capacity

In the timing-channel interpretation, the queue is a communication channel whose input is the arrival sequence and whose output is the departure sequence. A message $\lambda$4 is encoded into a causal arrival policy, and the decoder observes departure times. The rate of a code is $\lambda$5 bits per second, where $\lambda$6 upper-bounds the expected time by which the $\lambda$7-th departure occurs. At fixed output rate $\lambda$8, the $\lambda$9-packet block must finish by time $\mu$0 on average, yielding capacities $\mu$1 without feedback and $\mu$2 with feedback (Aptel et al., 2017).

For the continuous-time $\mu$3 specialization, the fixed-rate regime is

$\mu$4

A central formula is the feedback-capacity expression

$\mu$5

The paper recalls the Anantharam–Verdú 1996 result that, for exponential service, the corresponding upper bound on $\mu$6 is tight. Hence

$\mu$7

and after optimization over $\mu$8,

$\mu$9

In this sense, for the continuous-time $q\in(0,1]$0 timing channel, feedback does not increase capacity (Aptel et al., 2017).

The same paper emphasizes that this conclusion is special to the exponential case. It proves sufficient conditions under which feedback increases capacity for non-exponential FIFO service, including continuous-time uniform service on $q\in(0,1]$1, and gives an output-entropy-rate condition under which feedback does not increase capacity. The latter applies, for example, to single-server LCFS queues with bounded service times: $q\in(0,1]$2 A common misconception is therefore that “feedback helps” or “feedback does not help” is determined solely by the presence of one server. The paper instead shows that the answer depends on the service law, service discipline, and entropy properties of the output process (Aptel et al., 2017).

4. Weak feedback, ordinary feedback, and exponential memorylessness

The timing-channel literature distinguishes ordinary feedback from weak feedback. Ordinary feedback means that the encoder causally observes departure times $q\in(0,1]$3. Weak feedback means that the encoder instead observes the service-start times

$q\in(0,1]$4

Under FIFO,

$q\in(0,1]$5

The corresponding capacities satisfy

$q\in(0,1]$6

Weak feedback is central because it separates mere knowledge of service initiation from full knowledge of departure epochs (Aptel et al., 2017).

The weak-feedback upper bound is

$q\in(0,1]$7

This differs from the full-feedback optimization because $q\in(0,1]$8 is constrained to have the form

$q\in(0,1]$9

whereas under full feedback one optimizes over any $q$0 satisfying the mean constraint. This restriction is the mechanism by which feedback can become strictly useful for non-exponential FIFO service (Aptel et al., 2017).

For the $q$1 timing channel, however, exponential memorylessness neutralizes that extra leverage. The paper does not re-prove the original 1996 theorem, but it repeatedly identifies exponential service as the special case in which the no-feedback system can match the feedback-capacity bound. This suggests that in the exponential setting, once a packet begins service, residual uncertainty is statistically fresh in a way that makes causal departure information insufficient to enlarge the attainable information rate (Aptel et al., 2017).

The 2025 strategic paper provides a different but related use of exponential memorylessness. There, the system state for an arriving customer can be reduced to the number in the system, equivalently the joining position. That reduction underpins the threshold-strategy formulation and the finite-state QBD representation used to compute discounted sojourn transforms (Taylor et al., 15 Jul 2025).

5. Strategic joining, threshold equilibria, and ESS

In the observable Bernoulli-feedback queue, customers decide whether to join or balk after observing the current congestion level. In the 2021 model, customers are homogeneous with service reward $q$2 and waiting-cost rate $q$3. If $q$4 is the sojourn time, the payoff is

$q$5

The paper focuses on symmetric threshold strategies. For $q$6, with $q$7 and $q$8,

$q$9

Thus pure thresholds arise at integers and mixed thresholds at non-integers (Fackrell et al., 2021).

Let $1-q$0 denote the expected remaining sojourn time when the tagged customer is in position $1-q$1 and there are $1-q$2 customers in the system. The first-step analysis leads to the linear system

$1-q$3

that is, Poisson’s equation for an embedded discrete-time nonhomogeneous QBD. The paper proves that $1-q$4 is increasing in $1-q$5 and that $1-q$6 is increasing in the threshold $1-q$7. Consequently,

$1-q$8

is decreasing both in joining position and in the common threshold used by others (Fackrell et al., 2021).

The equilibrium structure is characterized by critical delay levels

$1-q$9

and, when $M/M/1$0, by the unique $M/M/1$1 satisfying

$M/M/1$2

The symmetric Nash equilibrium threshold is

$M/M/1$3

where $M/M/1$4. The abstract states that there exists a unique symmetric Nash equilibrium threshold strategy, with the usual knife-edge exception at $M/M/1$5 (Fackrell et al., 2021).

The same paper further proves that the symmetric equilibrium threshold is evolutionarily stable except in the indifference case $M/M/1$6. The ESS result uses the payoff functional $M/M/1$7 for a mutant threshold $M/M/1$8 in a population using threshold $M/M/1$9, and establishes the familiar separation between equilibrium best response and robustness to invasion (Fackrell et al., 2021).

6. Discounted rewards, reneging, and congestion paradoxes

The 2025 paper generalizes the payoff structure from linear waiting costs to a discounted reward model. A customer who joins receives payoff

$q$00

where $q$01 is a discount rate, $q$02 is an admission fee or outside-option cost, and $q$03 is the sojourn time until successful completion. The paper emphasizes three equivalent interpretations of $q$04: it is a discounted reward, the probability of completion before an independent exponential killing time, and the Laplace transform of the sojourn-time distribution (Taylor et al., 15 Jul 2025).

For the non-reneging case, the transformed values

$q$05

satisfy a discounted first-step recursion that can be written in matrix form as

$q$06

Because the tagged-customer chain is a finite-state continuous-time QBD, the paper solves this system by matrix-analytic methods following Dendievel–Latouche–Liu and Latouche–Ramaswami. It then numerically inverts the Laplace transform to recover the full cdf

$q$07

citing Kareem (2021) for inversion (Taylor et al., 15 Jul 2025).

The equilibrium characterization remains threshold-based. If

$q$08

then there exists a unique $q$09 such that

$q$10

The resulting symmetric Nash equilibrium threshold $q$11 has the same pure-versus-mixed structure as in the 2021 linear-cost model (Taylor et al., 15 Jul 2025).

Both the 2021 and 2025 papers analyze a reneging extension in which a customer whose service fails may choose to leave instead of rejoining, subject to the same threshold rule. The 2021 paper proves that the Nash equilibrium threshold with reneging is greater than or equal to that without reneging,

$q$12

and that for some parameter values the equilibrium expected payoff decreases despite the added flexibility. The 2025 paper establishes the same threshold comparison under discounted rewards and shows that if $q$13, then it can occur that

$q$14

This is one of the central congestion paradoxes of the strategic $q$15 feedback queue: options that make joining individually more attractive can induce more entry and lower stationary individual payoff in equilibrium (Fackrell et al., 2021, Taylor et al., 15 Jul 2025).

A second paradox in the discounted-reward model is that decreasing $q$16 or $q$17 can reduce the stationary individual payoff $q$18. The paper interprets lower $q$19 as less discounting and lower $q$20 as a cheaper outside option; both should increase willingness to join, and indeed they raise the equilibrium threshold, but the resulting congestion can dominate the direct gain (Taylor et al., 15 Jul 2025).

7. Methods, structural insights, and research significance

Across these papers, the $q$21 feedback queue serves as a tractable setting in which feedback creates endogenous dependence beyond the ordinary $q$22 countdown structure. In the Bernoulli-feedback queue, a tagged customer’s own service completion may recycle her to the back of the line, so delay analysis requires a two-dimensional state descriptor $q$23 rather than position alone. In the timing-channel model, feedback transforms the admissible arrival policies and motivates an optimization over induced waiting times $q$24 (Aptel et al., 2017).

Several structural themes recur. First, exponential service is analytically decisive. In timing capacity, it yields the no-feedback-gain identity $q$25. In strategic queueing, it supports Markovian state reduction and QBD methods. Second, threshold behavior is robust. It appears under linear waiting costs, discounted rewards, and deadline-type objectives derived from the numerically recovered sojourn distribution. Third, individual incentives and system welfare diverge. Lower waiting penalties, higher reward persistence, or reneging opportunities can all increase equilibrium entry while reducing equilibrium payoff (Fackrell et al., 2021, Taylor et al., 15 Jul 2025).

The broader significance of the recent literature is therefore not a single universal theorem about “feedback,” but a sharper taxonomy. In timing communication through an $q$26 queue, output feedback does not increase capacity because of exponential memorylessness. In observable strategic $q$27 queues with Bernoulli customer feedback, recirculation changes effective load from $q$28 to $q$29, generates threshold equilibria, and produces welfare reversals under both mean-based and distribution-sensitive objectives. A plausible implication is that the phrase “$q$30 feedback queue” should always be read with attention to which feedback mechanism is meant: informational feedback to the encoder, or physical feedback of customers to the tail of the queue (Aptel et al., 2017).