M / M / 1 Model With Unreliable Service and a Working Vacation

We derive an explicit closed form of the stationary distribution of an M/M/1 queue with unreliable service and a working vacation. We also show that the work in (Patterson & Korzeniowski, 2018) can be obtained as a special case of this model. Future work remains to be done; specifically, it may be possible to use the explicit stationary distribution given here to decompose the queue length into the sum of independent random variables. Consequently, it may then be possible to utilize Little’s Law (Little, 1961) to decompose the customer waiting time as well.


Introduction
Within the literature, a vacation queue is typically defined as a queue where the server is considered to be in a different state, generally defined to be a 'vacation' when the number of customers in the queue is below a certain threshold.For an example of a vacation queue within the M/G/1 framework, see (Levy & Yechiali, 1975).Working vacation queues typically refer to queues where the server utilizes its time on vacation for other purposes.For an example of a working vacation queue where the server uses its vacation to service the main queue, but at a reduced rate, see (Xu & Tian, 2009).
A queue with unreliable service is a queue where service may be unsuccessful any number of times before it is successful.This type of queue is important to study because it occurs naturally within a lot of systems.For example, imagine trying to have a conversation with someone in a quiet environment, such as a library-words spoken are generally heard and understood (i.e.service is rendered successfully every time).Now, imagine trying to have the same conversation in a noisy environment, such as a busy restaurant-it can be done, but you may need to repeat yourself (i.e.service may fail).Within the literature, this has been achieved two ways.First, using the M/G/1 framework, a M/PH/1 queue allows for a random number of Poisson distributed 'stages' within the service time.However, this method necessitates the restriction that: µ < β 1 + β 2 , where µ is the service rate, β 1 is the success rate, and β 2 is the failure rate (Latouche & Ramaswami, 1999).Second, one may use the framework of (Nuets, 1981) to construct a queue with identical stationary distribution to the M/PH/1 but does not inherit any restrictions on µ, β 1 or β 2 besides those necessary for positive recurrence, as seen in (Patterson & Korzeniowski, 2018).We extend the M/M/1 model with unreliable service defined in (Patterson & Korzeniowski, 2018) by including two service rates.The use of multiple service rates is important since the customer service time depends not only on the customer, as as it would be in the M/PH/1 queue, but on the state of the server at the time of service as well.This not only has the benefit of generalizing the results of (Patterson & Korzeniowski, 2018) by recovering this stationary distribution as a special case, but also cannot be recovered by previous results related to the M/PH/1.We adopt assumptions and terminology from (Patterson & Korzeniowski, 2018).Namely, service failure is not due to the server as it would be in breakdown models, nor due to the customer as it would be in some interruption models.Customer's do not leave the queue-that is we preserve the FCFS (First Come First Served) service protocol.We consider service failures to be due to external, random forces and repeat a customer's service until it has been completed successfully.Furthermore, neither the server nor customer know whether the service was successful until the service time has been completed, at which time we envision a 'quality check' to take place which determines if the service was a success or failure.

Definitions
We define our process, state space, and parameters as follows: Definition 2.1.Let {N(t) | t 0} be the number of customers in the queue at time t, 0 the server is on working vacation 1 the server is in a busy state and S(t) = { 1 immediately after service is rendered 0 otherwise Then {(N(t), J(t), S(t)) | t 0} is a Markov process on the state space: Define the following parameters: • λ : the rate of the Poisson arrivals process.
• µ b : the rate of service when the server is 'busy,' successful or not.
• µ v : the rate of service when the server is on 'vacation,' successful or not.
• β 1 : the rate of a successful service.
• β 2 : the rate of a failed service.
• θ : vacation duration is exponentially distributed with rate α.
Definition 2.2.We define the vacation policy: • When the server becomes idle (i.e.N(t) = 0), the server goes on a working vacation; by this we mean that customers arriving while the server is on vacation get served at a reduced rate µ v < µ b .• When the server is not idle (i.e.N(t) 0), a vacationing server begins a working vacation duration that is exponentially distributed with rate θ, after which it begins a busy period and operates at rate µ b until the server becomes idle again, renewing the process.• If a customer is served successfully while the server is on a working vacation and there are additional customers waiting in the queue, the server then immediately ends its vacation and enters into a busy state until the queue is emptied.
To help visualize this 3-dimensional Markovian process in 2-dimensions, we informally construct the state transition rate diagram in 2D.We define a 'successful service' similarly to that done in (Patterson & Korzeniowski, 2018) to be a transition from (n, j, 1) −→ (n − 1, 0|1, 0), which is represented in the state transition diagram as having rate β 1 .Accordingly, we will define a 'failed service' to be a transition from (n, j, 1) −→ (n, j, 0) with transition rate β 2 .We will compute the probabilities of a 'successful' or 'failed' service in an explicit manner by considering the transition probabilities of the embedded Markov Chain and will note a similarity with the results from (Patterson & Korzeniowski, 2018).

Infinitesimal Matrix
where

The Quadratic Matrix Equation
Thanks to Lemma 4.1 from (Patterson & Korzeniowski, 2018), which is based on the framework developed by (Neuts, 1981), we seek the minimal non-negative solution R to the quadratic matrix equation: We will again employ the direct method whereby we solve the system of equations generated by equating the matrices entry by entry.
r 11 r 12 r 13 r 14 r 21 r 22 r 23 r 24 r 31 r 32 r 33 r 34 r 41 r 42 r 43 r 44     =⇒ (2) can be restated as the following system: The analytical minimal non-negative solution to (3) is given by:

The Spectral Radius of R
We compute the spectral radius of R explicitly and show that the sufficient condition under which our model will be positive recurrent has not changed from the case in (Patterson & Korzeniowski, 2018).
Corollary.By Lemma 4.1 from (Patterson & Korzeniowski, 2018), the infinitesimal matrix Q given in equation ( 1) is positive recurrent if and only if: Proof.The spectral radius of R is computed by solving the scalar quadratic equation generated by det(R − ρ i I) = 0, yielding that {ρ i } i=0,1,2,3 satisfies the following quadratic equations: By inspection, the largest of these eigenvalues in ( 5) and ( 6) will contain the positive radicals.Next we show that ρ 0 ρ 2 .

The Explicit Form of R k
To find an expression for R k , we utilize the block upper-triangular structure of the matrix R given in (4).To that end, we prove the following: Proof.
, and write R k+1 as follows: )) )) , we find: We use Mathematical Induction by noting that: and assume the result for A k , write A k+1 : Remark.Three substitutions were needed in this derivation; namely: , and These can readily be verified from (6). where: ] , where: ) Proof.The above result is the consequence of previous works, specifically Lemma 6.1, Proposition 6.2, and the observation that B is entry-wise identical to the matrix R from (Patterson & Korzeniowski, 2018) after letting µ = µ b .The rest is merely the computation of C(k) = ∑ k−1 i=0 A i CB k−i−1 which is tedious but straightforward.

Figure 1 .
Figure 1.3D Markovian state transition rate diagram in 2D Proposition 6.3.Using the block-matrix form of R =