Question

Consider an infinitely many-server queue with an exponential service time distribution with parameter \(\mu\). Suppose customers arrive in batches with the interarrival time following an exponential distribution with parameter \(\lambda\). The number of arrivals in each batch is assumed to follow the geometric distribution with parameter \(\rho(0<\rho<1)\), i.e., Pr \(\\{\) number of arrivals in a batch has size \(k\\}\) \(=\rho^{k-1}(1-\rho)(k=1,2, \ldots)\) Formulate this process as a continuous time Markov chain and determine explicitly the infinitesimal matrix of the process.

Short answer

To model the given queuing system as a continuous-time Markov chain, we define the states to represent the number of customers in the system and the transition rates as the rate of customer arrival and departure. For customer arrival in batches, we have the rate of arrival of a batch as \(q_{ij}= \lambda\rho^{k-1}(1-\rho)\) for a batch of size \(k\), and the departure rate for each customer as \(q_{ij} = i\mu\). The infinitesimal matrix \(Q\) can then be formulated as: \[Q = \begin{pmatrix} -\lambda(1-\rho) & \lambda(1-\rho) & \lambda\rho(1-\rho) & \lambda\rho^2(1-\rho) & \cdots \\ \mu & -(1\mu+\lambda(1-\rho)) & \lambda(1-\rho) & \lambda\rho(1-\rho) & \cdots \\ 0 & 2\mu & -(2\mu+\lambda(1-\rho)) & \lambda(1-\rho) & \cdots \\ \vdots & \vdots & \vdots & \vdots & \ddots \end{pmatrix}\]

Step-by-step solution

Define the States of the Markov chain

Let \(X(t)\) denote the number of customers in the system at time \(t\). For our Markov chain, the state space is \(\{0, 1, 2, ...\}\), and each state corresponds to the number of customers in the system.

Define Transition Rates of the Markov chain

Now, we need to define the transition rates \(q_{ij}\), which represent the rate of transition from state \(i\) to state \(j\). There are two types of events that can change the number of customers in the system: customer arrivals (shown as a batch) and customer departures (assumed to be exponential). Customer arrival: Batches arrive with an exponential distribution at rate \(\lambda\), with the batch size following the geometric distribution specified in the exercise. This means the rate of arrival of a batch is \(\lambda\rho^{k-1}(1-\rho)\) for a batch of size \(k\). Customer departure: Since there are infinitely many servers in the system, every customer has their own server. The service time has an exponential distribution with parameter \(\mu\). Therefore, the departure rates for each customer (which will lead to state transitions) are \(\mu\). Now let's define the transition rates \(q_{ij}\): 1. For the arrival of a batch of size \(k\) (transitioning from state \(i\) to state \(i+k\)), we have: \(q_{ij}= \lambda\rho^{k-1}(1-\rho)\) when \(j=i+k\) 2. For the departure of one customer (transitioning from state \(i\) to state \(i-1\)), we have: \(q_{ij} = i\mu\) when \(j=i-1\) 3. For all other transitions, \(q_{ij} = 0\)

Build the Infinitesimal Matrix

Now, we will build the infinitesimal matrix \(Q\) for our Markov chain based on the transition rates we have determined. The elements of the infinitesimal matrix are defined as follows: \[Q(i, j)=\begin{cases} - q_{ii} = -\sum_{k\neq i} q_{ik}, & \text{for }i=j \\ q_{ij}, & \text{for }i\neq j \end{cases}\] Thus, the infinitesimal matrix takes the form: \[Q = \begin{pmatrix} -\lambda(1-\rho) & \lambda(1-\rho) & \lambda\rho(1-\rho) & \lambda\rho^2(1-\rho) & \cdots \\ \mu & -(1\mu+\lambda(1-\rho)) & \lambda(1-\rho) & \lambda\rho(1-\rho) & \cdots \\ 0 & 2\mu & -(2\mu+\lambda(1-\rho)) & \lambda(1-\rho) & \cdots \\ \vdots & \vdots & \vdots & \vdots & \ddots \end{pmatrix}\] This is the infinitesimal matrix of the continuous-time Markov chain representing our queuing system.

Key concepts

Exponential Service Time Distribution

The exponential service time distribution is a key concept in the analysis of continuous-time Markov chains, particularly in queuing theory where it is used to model the time between consecutive service completions. The parameter \( \mu \) represents the rate at which services are completed, and it defines the average length of time a service takes. In mathematical terms, the probability that a service time is greater than some value \( t \) can be described by the exponential distribution function:
\[ P(Service\ Time > t) = e^{-\mu t} \]
One unique characteristic of the exponential distribution is the memoryless property, which means the probability of an event occurring in the future is independent of how much time has already elapsed. This property simplifies the formulation of Markov processes, as transitions depend only on the current state and not on the history of the system. Understanding this distribution is essential when dealing with systems where the 'exponential assumption' about service times is made, as is the case in our exercise scenario with an infinitely many-server queue.

Batch Arrivals

Batch arrivals refer to a situation in a queuing system where multiple customers or items arrive simultaneously and are served. This concept contrasts with individual or 'single' arrivals and incorporates a more complex arrival process into the modeling of systems. In the context of the exercise, customers are arriving in groups with the time between each group's arrival following an exponential distribution with the rate parameter \( \lambda \). This rate parameter governs the likelihood of a batch arrival occurring during a tiny interval of time.

Applying batch arrivals to the continuous-time Markov chain significantly alters the state transition dynamics because a single arrival event can now increase the number of customers in the system by more than one. The process of arrival in batches can be seen in many real-world situations like packets arriving in networks, bulk purchases in inventory systems, or people arriving in groups at a service facility. Understanding batch arrivals helps in creating models that better reflect the complexity of real-world systems and aid in designing more efficient service mechanisms to handle simultaneous demand.

Geometric Distribution

The geometric distribution is applied in our exercise to model the number of customers in each arriving batch. A key property of the geometric distribution is that it models the number of Bernoulli trials needed to get one success, with \( \rho \) representing the success probability. In the context of our queuing system, this translates to the probability of having a batch of exactly \( k \) customers being \( \rho^{k-1}(1-\rho) \) with \( k=1,2,3,... \)

The geometric distribution is discrete and supports the memoryless property, similar to the exponential distribution, meaning the probability of success in further trials is unaffected by past failures. It has practical relevance in scenarios where the number of attempts before a success must be taken into account, such as the size of groups (batches) arriving at a service point. By incorporating the geometric distribution into the modeling of our queuing system, we acknowledge the variability of group sizes, which is crucial for accurately simulating and understanding service dynamics and requirements.