Bayesian analysis of an M |M |1|∞ queueing model

In this paper, by considering an M |M |1|∞ queueing model, Bayesian estimation of traffic intensity and measures of system performance are worked out under the squared error loss function (SELF) based on the observed data on independent interarrival and service times. Further, minimum posterior risk associated with Bayes estimators of traffic intensity and system performance measures are obtained under SELF. Numerical illustration of the performance of the estimates is given through simulation study. It is shown that Bayes estimators perform better than the maximum likelihood estimators under the influence of prior informations.


Introduction
Queueing models are often used in the design and analysis of telecommunication systems, traffic systems, service systems and so on. Since the arrival time and service time of entities in the queue are stochastic, it will be of interest to carry out inferential procedures to study and analyze the behaviour of the parameters of the queueing models by assuming suitable probability distributions. This can be done either through the frequentist or Bayesian approach. In case where the probability distribution of either or both of the arrival and service times are not known, non-parametric inferential procedures can be employed to analyze the queueing model; see for example Schweer and Wichelhaus (2015). In this paper, we take a parametric approach by assuming exponential distributions for arrival and service times. A detailed survey on different inferential procedures and its applications to various queueing models can be found in Asanjarani et al. (2021). Among many queueing models available in the literature, M |M |1|∞ model has received more attention primarily due to less model complexities. This model assumes only one service station and does not put a cap on the queue size. The main purpose of this article is to apply certain statistical inference procedures for an M |M |1|∞ queueing model with Poisson input and exponential service times from a Bayesian perspective. It is often the case that information is available on the parameters of the interarrival or service time distribution from prior experiments or from prior analysis of the interarrival or service time data. Bayesian approach provides the methodology by incorporating prior information to the current data.
Maximum likelihood estimator (MLE) for measures of system performance in the case of M |M |1 queueing model is discussed in detail in Sharma and Kumar (1999), Yadavalli et al. (2004), Srinivas et al. (2011), Chowdhury (2005, 2010), Choudhury and Borthakur (2008), and Mukherjee (2011, 2013). All these authors have studied M |M |1 queueing models based on the number of customers present on n iid M |M |1 queueing systems. Choudhury and Borthakur (2008) and Chowdhury (2005, 2010) have also studied Bayes estimation based on the number of customers present in the system. However, in this paper, we consider an M |M |1 queueing model based on observed data on interarrival and service times.
Throughout the paper, we consider the queues in which there are m interarrival times and n service times. For the queueing system under consideration, we use appropriate prior distributions for the parameters and evaluate the posterior distributions along with Bayes estimators and minimum posterior risks of the estimators. The rest of the paper is organized as follows: In Section 2, we introduce the model and describe the inferential aspects including Bayes estimators of the parameters and measures of system performance under SELF. Numerical illustration based on simulation study is presented in Section 3. Conclusion of the paper is given in Section 4.

System description of M |M |1|∞ queue and its performance measures
Assume that arrivals follow Poisson process with mean λt. In other words, interarrival times are independent and follow an exponential distribution with mean 1 λ . i.e., For service times, we assume exponential distribution with mean 1 µ . The steady state distribution of the number of entities present in the M |M |1|∞ queueing system is given by see Shortle et al. (2018). Let ρ = λ µ , then (2) can be written as The number of entities in the system (L S ), number of entities in the queue (L Q ) and the probability that there are at least k entities in the queue denoted by (Q(k)) are given by Similarly, let y 1 , y 2 , . . . , y n be a random sample of size n, where y j is j-th observed service time, j = 1, 2, . . . , n. Also let u = m i=1 x i and v = n j=1 y j . Since maximum likelihood estimators of λ and µ, subject to λ < µ are given byλ = 1 x = m u andμ = 1 y = n v , wherex andȳ are the sample means of observed interarrival times and service times respectively, the MLE of ρ is given byρ = min(1,λ µ ) = min (1, mv nu ). In the subsequent sections, Bayes estimators of traffic intensity and measures of system performance are derived under SELF.
Bayesian analysis of an M |M |1|∞ queueing model

Bayes estimator of λ under SELF
Given the data ( x, y), the Bayes estimator of λ under SELF denoted asλ B is obtained aŝ From (13), Hence, from (12), we get,λ

Bayes estimator of µ under SELF
The Bayes estimator of µ under SELF denoted byμ B is obtained aŝ

Bayes estimator of ρ under SELF
Bayes estimator of ρ under SELF denoted byρ B is given bŷ

Bayes estimators of measures of system performance
In this section, Bayes estimators of different queueing performance measures are derived in the steady state. More specifically, we estimate measures of system performance namely, the expected number of entities in the system, average queue length, mean sojourn time and the probability of minimum queue size by observing the arrival and service times of entities.

Bayes estimator of L S under SELF
Bayes estimator of L S under SELF denoted byL S B is obtained aŝ

Bayes estimator of L Q under SELF
Bayes estimator of average queue length L Q under SELF denoted byL Q B is given bŷ

Bayes estimator of Q(k) under SELF
Bayes estimator of probability of minimum queue size Q(k) under SELF denoted byQ (k) B is obtained aŝ

Minimum posterior risks of the parameters and system performance measures
In this section, we derive the minimum posterior risks of the parameters λ, µ and ρ and system performance measures L S , L Q and Q(k).

Minimum posterior risk of λ
By definition, Bayesian analysis of an M |M |1|∞ queueing model Also, Therefore, It can be shown that Hence, Thus, Using (22) and (15) in (21) and simplifying it, we get

Minimum posterior risk of ρ
The minimum posterior risk of the traffic intensity parameter ρ is given by

Minimum posterior risks of system performance measures
By proceeding as explained in the previous sub sections, the expressions for minimum posterior risks of L S , L Q and Q(k) are derived as follows:  In Tables 1-6, besides ρ, we also compare the Bayes and ML estimators of the interarrival time parameter λ, the service time parameter µ, the mean number of entities in the system L S , and the mean number of entities in the queue L Q for different sets of parameters. Note that the bias and the MSEs of ML estimators of L S and L Q are not presented in the tables because they are not computable due to the fact that the ML estimator of ρ have value 1 for some simulated samples; see equations (4) and (5). Note that ML estimator ρ M L = 1 whenx ≤ȳ.

Numerical illustration and the choice of the prior
We consider different sets of sample sizes (m, n) ∈ {(10, 15), (20, 30), (15, 10), (30, 20)}. We vary the hyperparameters (a, b) of the distribution of ρ so that one set of (a, b) yields ρ(= λ µ ) close to E[ρ](= a (a+b) ), one set yields ρ away from E[ρ], and one set matches with the non-informative prior U [0, 1]. The reason of doing so was that, in practice, an expert opinion or empirical estimates of (a, b) may be based on the average of the past history or past data but the actual ρ may or may not be close to E[ρ].
Tables 1-6 show that the absolute mean bias of the Bayes estimatorsλ B andμ B are smaller compared to those of the ML estimators when E[ρ] is closer to ρ, but larger when it is further away from ρ. The bias of the ML estimator of ρ is slightly better than the bias of the Bayes estimator for all prior choices. In terms of the mean square errors, when actual ρ is closer to zero, Bayes estimators ofλ B ,μ B andρ B perform better under the prior when E[ρ] ≈ ρ but not as good as when E[ρ] is further away from ρ. However, when the actual ρ is near 0.5, thenλ B ,μ B andρ B are better than the ML estimators under the prior when E[ρ] = 0.5 (non-informative prior) or when E[ρ] < 0.5. When the actual ρ > 0.5,λ B ,μ B andρ B perform better under both noninformative prior or when E[ρ] ≈ ρ.
In summary, the simulation results show that when ρ is closer to zero, ML estimators are overall superior. However whenever ρ is near 0.5 or higher, the Bayes estimators are superior under non-informative priors or when E[ρ] ≈ ρ. For the estimators of L S and L Q , the ML estimators are not comparable, but the Bayes estimators produce reasonably good results in terms of both bias and MSE under non-informative prior or when the prior is such that E[ρ] is near 0.  In general, for Bayes estimation, our recommendation is to choose the prior (8) such that the prior on ρ is non-informative, i.e., a = 1, b = 1, and to choose prior of µ to be Gamma distribution with hyperparameters ω and δ which can be based on expert opinion. ω and δ can also be estimated from the marginal likelihood of (y 1 , y 2 , . . . , y n ). Another choice is to estimate the hyperparameters of (8) using the marginal likelihood of (x 1 , x 2 , . . . , x m ) and (y 1 , y 2 , . . . , y n ) as we illustrate below.

Estimation of hyperparameters
The hyperparameter vector θ = (a, b, δ, ω) can be estimated by maximizing the marginal likelihood. The marginal likelihood is given by The parameter vector θ can be estimated by solving the system of equations ∂LogL ∂ θ = 0. However, the system of equations will be highly non-linear due to the presence of gamma functions involved in the integration. The issue of integration can be resolved by using EM algorithm, but the system of equations will still be complex due to the gamma function.Thus, we propose, the method of moments estimator that involve a tractable system of equations.
Note that the method of moments estimators are not as efficient as the marginal ML estimators. The first two moments of the distributions of X and Y are given by . Now by replacing E(X), E(X 2 ), E(Y ) and E(Y 2 ) in the above equations by its corresponding sample moments m 1X , m 2X , m 1Y and m 2Y and solving for a, b, ω and δ, we get the moment estimators. After solving for a, b, ω and δ, we get the following expressions for the estimators of a, b, ω and δ.

Conclusion
An attempt is made in this paper to study the Bayesian estimation of M |M |1|∞ queueing model when sample observations are available on independent interarrival and service times of entities at some points of times. Bayes estimators of traffic intensity and system performance measures have been derived assuming squared error loss function. Bayes estimators, studied in the past, assume available observations on the queue sizes which are sometimes not practical. We obtain Bayes estimators when observations are available on arrival times and service times which are sometimes easy to document. These estimators show a superior performance for the most part against the ML estimators. A part of deficiency in the ML estimators is due to the constraint λ < µ. In this case, the ML estimates of L S and L Q does not even exist ifx ≤ȳ, whereas, Bayes estimates can be obtained.