Abstract
The paper deals with the asymptotic joint posterior distribution of \((\theta , \phi )\) in a GI / G / 1 queueing system over a continuous time interval (0, T] where \(\theta \) and \(\phi \) are unknown parameters of arrival process and departure process respectively and T is a suitable stopping time.
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1 Introduction
Though statistical inference plays a major role in any use of queueing models, study of asymptotic inference problems for queueing system can be hardly traced back to the works by Basawa and Prabhu (1981, 1988) where they have discussed about the maximum likelihood (ML) estimators of the parameters in single server queues. Basawa et al. (1996) have studied the consistency and asymptotic normality of the parameters in a GI / G / 1 queue based on information on waiting times. Acharya (1999) has studied the rate of convergence of the distribution of the maximum likelihood estimators of the arrival and the service rates from a single server queue. Acharya and Mishra (2007) have proved the Bernstein–von Mises theorem for the arrival process in a M / M / 1 queue.
From a Bayesian outlook, inferences about the parameter are based on its posterior distribution. The study of asymptotic posterior normality can be traced back to the time of Laplace and it has attracted the attention of many authors. A conventional approach to such problems starts from a Taylor series expansion of the log-likelihood function around the maximum likelihood estimator (MLE) and proceeds from there to develop expansions that have standard normal as a leading term and hold in probability or almost surely, given the data. This type of study have not been done in queueing system. For the general set up in this direction the previous work seems to be those by Walker (1969), Johnston (1970) for i.i.d observations; Hyde and Johnston (1979), Basawa and Prakasa Rao (1980), Chen (1985) and Sweeting and Adekola (1987) for stochastic process. The most recent work was done by Kim (1998) in which he provided a set of conditions to prove the asymptotic normality under quite general situations of possible non-stationary time series model and Weng and Tsai (2008) where they studied asymptotic normality for multiparameter problems.
In this paper, our aim is to prove that the joint posterior distribution of \((\theta , \phi )\) is asymptotically normal for GI / G / 1 queueing model in the context of exponential families. In Sect. 2 we introduce the model of our interest and explain some elements of maximum likelihood estimator (MLE) as well as Bayesian procedure. In Sect. 3 we prove our main result. For the illustration purpose we provide an example Sect. 4. Section 5 deals with the simulation study while in Sect. 6 concluding remarks are given.
2 GI / G / 1 Queueing Model
Consider a single server queueing system in which the interarrival times \(\{u_{k}, k\ge 1\}\) and the service times \(\{v_{k}, k\ge 1\}\) are two independent sequences of independent and identically distributed nonnegative random variables with densities \(f(u; \theta )\) and \(g(v; \phi )\), respectively, where \(\theta \) and \(\phi \) are unknown parameters. Let us assume that f and g belong to the continuous exponential families given by
and
where \(\Theta _1=\{\theta >0:~ k_1(\theta )< \infty \}\) and \(\Theta _2=\{\phi >0:~ k_2(\phi )< \infty \}\) are open subsets of \(\mathbb {R}\). It is easy to see that, \(E_{\theta }(h_1(u)) = k_1^{\prime }(\theta )\), \(var_{\theta }(h_1(u))=k_1^{\prime \prime }(\theta )\), \(E_{\phi }(h_2(v)) = k_2^{\prime }(\phi )\), \(var_{\phi }(h_2(v)) = k_2^{\prime \prime }(\phi )\), are supposed to be finite.
For simplicity we assume that the initial customer arrives at time \(t=0\). Our sampling scheme is to observe the system over a continuous time interval (0, T], where T is a suitable stopping time. The sample data consist of
where A(T) is the number of arrivals and D(T) is the number of departures during (0, T]. Obviously no arrivals occur during \([\sum _{i=1}^{A(T)} u_{i}, T]\) and no departures during \([\gamma (T)+\sum _{i=1}^{D(T)}v_{i}, T]\), where \(\gamma (T)\) is the total idle period in (0, T].
The likelihood function based on data (2.3) is given by
where F and G are distribution functions corresponding to the densities f and g respectively.
The approximate likelihood \(L_{T}^{(a)}(\theta ,\phi )\) is defined as
where
and
The maximum likelihood estimates obtained from (2.5) are asymptotically equivalent to those obtained from (2.4) provided that the following two conditions are satisfied for \(T \rightarrow \infty \):
and
The implications of these conditions have been explained by Basawa and Prabhu (1988).
Basawa and Prabhu (1988) have shown that the maximum likelihood estimator of \(\theta \) and \(\phi \) are given by
where \(\eta _i^{-1}(.)\) denotes the inverse functions of \(\eta _i(.)\) for \(i=1, 2\) and
and
The Fisher information matrix is given by
Under suitable stability conditions on stopping times, Basawa and Prabhu (1988) have proved that the estimators \(\hat{\theta }_T\) and \(\hat{\phi }_T\) are consistent, i.e,
and
where \(\theta _0\) and \(\phi _0\) denote the true value of \(\theta \) and \(\phi \) respectively, and the symbol \(\Rightarrow \) denotes the convergence in distribution.
From Eq. (2.5) we have the loglikelihood function
where
and
Let
Similarly \(\ell _T^{'}(\hat{\theta }_T)\), \(\ell _T^{'}(\hat{\phi }_T)\), \(\ell _T^{'}(\phi _0)\), \(\ell _T^{''}(\phi _0)\), \(\ell _T^{''}(\hat{\theta }_T)\) and \(\ell _T^{''}(\hat{\phi }_T)\) are defined.
Let \(\pi _1(\theta )\) and \(\pi _2(\phi )\) be the prior distributions of \(\theta \) and \(\phi \) respectively. Let the joint prior distribution \(\theta \) and \(\phi \) be \(\pi (\theta , \phi )\). Since the interarrival time and service time distributions are independent, so we have \(\pi (\theta , \phi )=\pi _1(\theta ) \pi _2(\phi )\). Then the joint posterior density of \((\theta , \phi )\) is
with
and
the marginal posterior densities of \(\theta \) and \(\phi \), respectively. Let \(\tilde{\theta }_T\) and \(\tilde{\phi }_T\) be Bayes estimator of \(\theta \) and \(\phi \) respectively.
In the next section we will state and prove our main result.
3 Main Result
Theorem 3.1
Let \((\theta _0, \phi _0)\in \Theta _1 \times \Theta _2\). If the prior densities \(\pi _1(\theta )\) and \(\pi _2(\phi )\) are continuous and positive at \(\theta _0\) and \(\phi _0\) respectively then, for any \(\alpha _i\), \(\beta _i\) such that \(-\infty \le \alpha _i \le \beta _i \le \infty \), \(i=1, 2\), the posterior probability that \((\hat{\theta }_T + \alpha _1 \sigma _T \le \theta \le \hat{\theta }_T + \beta _1 \sigma _T, \hat{\phi }_T + \alpha _2 \tau _T \le \phi \le \hat{\phi }_T + \beta _2 \tau _T)\), namely
tends in \([P_{(\theta _0, \phi _0)}]\) probability to
as \(T\rightarrow \infty \), where \(\sigma _T\) and \(\tau _T\) are the positive square roots of \([-\ell _T^{''}(\hat{\theta }_T)]^{-1}\) and \([-\ell _T^{''}(\hat{\phi }_T)]^{-1}\) respectively.
Proof of Theorem 3.1
The former integral of the above theorem can be written as the product of the integrals of the marginal posterior densities, i.e.,
and the convergence of both can be established separately.
For any \(\delta >0\), let us write \(\mathcal {N}(\varrho , \delta )=(\varrho -\delta , \varrho +\delta ) \) with \(\varrho \in \Theta _1\) and \(\mathcal {J}_B=\int _{B}L_T^{(a)}(\theta ) \pi _1(\theta ) d\theta \) where \( B \subseteq \Theta _1\). Hence,
with \(\delta _T=\frac{\sigma _T(\beta _1-\alpha _1)}{2}\) and \(\theta _T=\hat{\theta }_T +\frac{\sigma _T(\alpha _1+\beta _1)}{2}\). Then, we want to prove that
in probability \([P_{\theta _0}]\), where \(\Phi (z)=\frac{1}{\sqrt{2\pi }} \int _{-\infty }^{z}e^{-\frac{s^2}{2}}ds\).
Let us split \(\mathcal {J}_{\Theta _1}\) into \(J_{\Theta _1 \setminus \mathcal {J}_{\mathcal {N}(\theta _0, \delta )}}\) and \(\mathcal {J}_{\mathcal {N}(\theta _0, \delta )}\). Then, to obtain the above result it is sufficient to prove that the following statements holds in probability \([P_{\theta _0}]\) : For some \(\delta >0\),
-
(a)
\( \lim \limits _{ T \rightarrow \infty } [L_T^{(a)}(\hat{\theta }_T) \sigma _T]^{-1} J_{\Theta _1 \setminus \mathcal {J}_{\mathcal {N}(\theta _0, \delta )}} = 0 \)
-
(b)
\( \lim \limits _{ T \rightarrow \infty } [L_T^{(a)}(\hat{\theta }_T) \sigma _T]^{-1} \mathcal {J}_{\mathcal {N}(\theta _0, \delta )} = (2\pi )^\frac{1}{2} \pi _1(\theta _0) \)
-
(c)
\( \lim \limits _{ T \rightarrow \infty } [L_T^{(a)}(\hat{\theta }_T) \sigma _T]^{-1} \mathcal {J}_{\mathcal {N}(\theta _T, \delta _T)} = (2\pi )^\frac{1}{2} \pi _1(\theta _0) (\Phi (\beta _1)-\Phi (\alpha _1)) \)
Define
If \(\theta \) belongs to \(\mathcal {N}(\theta _0,\delta )\) for some \(\delta >0\), \(\ell _T^{\prime \prime }(\theta )/ \ell _T^{\prime \prime }(\theta _0)\) is close enough to 1 and, since \(\hat{\theta }_T \rightarrow \theta _0\) almost surely, \( \ell _T^{\prime \prime }(\hat{\theta }_T)/ \ell _T^{\prime \prime }(\theta _0)\) is almost surely close to 1 for T sufficiently large. Therefore we can deduce that for given \(\varepsilon >0\), we can take \(\delta \) such that, if T is large enough,
Consider also
Since \(\ell _T(.)\) has a strict maximum at \(\hat{\theta }_T\), it is obvious that \(q_T(.)\) is negative on \(\Theta _1 \setminus \mathcal {N}(\theta _0,\delta )\) for T large enough. Moreover, since \(\hat{\theta }_T \rightarrow \theta _0\) almost surely, it can be shown that there exists a positive constant \(\kappa (\delta )\) such that
Now,
We have \(-\ell _T^{\prime \prime }(\theta _0)= A(T) \sigma ^2(\theta _0)\) diverges to \(\infty \) almost surely as \(T \rightarrow \infty \). So, in the above expression
in probability and, using Eq. (3.5), for some constant M and T large enough
in probability and, consequently (a) holds.
Let us prove (b). Write
Using Taylor expansion around \(\hat{\theta }_T\),
for \(\bar{\theta }_T=\theta +\xi (\hat{\theta }_T-\theta )\) with \(0<\xi <1\). Thus letting
we have
Using Eqs. (3.8) and (3.9) in Eq. (3.7) and, for some \(\delta >0\) and T large enough such that \(\hat{\theta }_T \in \mathcal {N}(\theta _0,\delta )\), we have, for every \(\theta \in \mathcal {N}(\theta _0,\delta )\)
and consequently,
Since \(\pi _1(\theta )\) is continuous and positive at \(\theta =\theta _0\), then for given \(0<\varepsilon <1\), we can choose \(\delta \) small enough so that
Denote
Then from Eq. (3.12) we get that
If \(\sup _{\theta \in \mathcal {N}(\theta _0,\delta )} |R_T|< \varepsilon <1\), then
and for \(\eta =+\varepsilon \) or \(-\varepsilon \), making a change of variable,
Since \(\sigma _T^{-1} \rightarrow \infty \) and \(\hat{\theta }_T \rightarrow \theta _0\) almost surely, it is deduced that the limits \((\theta _0-\delta -\hat{\theta }_T)(1+\eta )^\frac{1}{2}\sigma _T^{-1}\) and \((\theta _0+\delta -\hat{\theta }_T)(1+\eta )^\frac{1}{2}\sigma _T^{-1}\) of the integrals in the above equation converges to \(-\infty \) and \(\infty \) respectively. Therefore, the term in square brackets in Eq. (3.14) converges to 1. Thus, using an appropriate bound on \(R_T\) it follows that,
in probability as \(T \rightarrow \infty \) and, using the above expression with the Eq. (3.13) we have the following bounds for \(\mathcal {J}_{\mathcal {N}(\theta _0,\delta )}\):
Hence (b) holds.
Finally, let us show (c). Using the same arguments and notations above, given \(\varepsilon >0\), there exists \(\delta \) such that if \(\mathcal {N}(\theta _T,\delta _T) \subseteq \mathcal {N}(\theta _0,\delta )\) for T large enough then
While the last term in Eq. (3.14) becomes
Therefore, we obtain that
and now (3.3) is established.
Similarly, using the same arguments as in the above, it can be shown that
in probability \([P_{\phi _0}]\) and the proof is completed. \(\square \)
4 Example
Let us consider a M / M / 1 queueing system. Under the Markovian set-up we have
So, the loglikelihood function is written as
and the MLEs are given by
Here \(\sigma _T=\left[ -\ell _T^{''}(\hat{\theta }_T)\right] ^{-\frac{1}{2}}=\frac{\sum _{i=1}^{A(T)}u_i}{\sqrt{A(T)}}\) and \(\tau _T=\left[ -\ell _T^{''}(\hat{\phi }_T)\right] ^{-\frac{1}{2}} =\frac{\sum _{i=1}^{D(T)}v_i}{\sqrt{D(T)}}\).
Let us assume that the conjugate prior distributions of \(\theta \) and \(\phi \) are gamma distributions with hyper-parameters \((a_1, b_1)\) and \((a_2, b_2)\), that is
where \(a_i, b_i > 0\) for \(i=1,2\).
Then, the posterior distribution of \(\theta \) can be computed as:
Similarly,
It is easy to see that
Here, the posterior distributions of \(\theta \) and \(\phi \) are seen to be gamma distributions [\(\text {Gamma}(A(T)+a_1, \sum _{i=1}^{A(T)}u_i+b_1)\) and \(\text {Gamma}(D(T)+a_2, \sum _{i=1}^{D(T)}v_i+b_2)\)]. Hence, by Central Limit Theorem (CLT), the joint posterior distribution converges to normal distribution as \(T \rightarrow \infty \).
5 Simulation
For the feasibility of the main result discussed in Sect. 3, simulation was conducted for M / M / 1 queueing system. For given values of true parameters \(\theta _0\) and \(\phi _0\) MLEs (\(\hat{\theta }_T\) and \(\hat{\phi }_T\)) are computed at different time interval (0, T]. Also by choosing different values of hyper-parameters of gamma distribution we compute the Bayes estimators (\(\tilde{\theta }_T\) and \(\tilde{\phi }_T\)) of \(\theta \) and \(\phi \). Here, we consider two pair of true value of parameters \(\theta _0\) and \(\phi _0\) as (1, 2) and (2, 3). For the hyper-parameters we have taken as: \((a_1, b_1)=(1.5, 2.5)\), \((a_2, b_2)=(3, 3.5)\) and \((a_1, b_1)=(3, 5)\), \((a_2, b_2)=(4, 5.5)\). The simulation procedure are repeated 10000 time to estimate the parameters. The computed values of estimators and their respective standard errors are presented in Tables 1, 2 and 3. The values in the parenthesis indicate the standard errors.
6 Concluding Remarks
In simulation study we present the estimates by proposed methods. It is clear that the estimators are quite closer to the true parameter values and their standard errors are negligible.
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The authors are thankful to the referees for their comments and useful suggestions.
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Singh, S.K., Acharya, S.K. Normal Approximation of Posterior Distribution in GI / G / 1 Queue. J Indian Soc Probab Stat 20, 51–64 (2019). https://doi.org/10.1007/s41096-018-0055-y
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DOI: https://doi.org/10.1007/s41096-018-0055-y
Keywords
- GI / G / 1 queue
- Exponential families
- Maximum likelihood estimator
- Posterior distribution
- Asymptotic normality