Programmers at Prolific Programming Unlimited connect to a...

Programmers at Prolific Programming Unlimited connect to a timesharing system over dedicated dial-up communication trunks. Arrivals of incoming programmer calls to the trunks can be modeled as a Poisson arrival process with the mean rate of 0.2 calls per minute. The length of each time-sharing session can be modeled as a uniformly distributed random variable in the interval 10 to 50 minutes; session length is independent of the load. (a) How many dial-up lines (trunks) should there be to make sure that less than 10 percent of incoming programmer calls get busy signals? (b) With this number of lines, what is the mean number of timesharing sessions in progress? (c) What is the probability that 6 or more of the lines are in use?

Programmers at Prolific Programming Unlimited connect to a timesharing system over dedicated dial-up communication trunks. Arrivals of incoming programmer calls to the trunks can be modeled as a Poisson arrival process with the mean rate of 0.2 calls per minute. The length of each time-sharing session can be modeled as a uniformly distributed random variable in the interval 10 to 50 minutes; session length is independent of the load.
(a) How many dial-up lines (trunks) should there be to make sure that less than 10 percent of incoming programmer calls get busy signals?
(b) With this number of lines, what is the mean number of timesharing sessions in progress?
(c) What is the probability that 6 or more of the lines are in use?

Key Concepts

Poisson Process

A Poisson process is a stochastic process used to model random events that occur independently and with a constant average rate over time. In the context of telecommunications or network systems, it is commonly used to represent the arrival of calls or requests. The key property is that the number of events occurring in disjoint time intervals are independent, and the probability of a single event occurring in a very small time interval is proportional to the length of that interval.

Uniform Distribution

A uniform distribution is a probability distribution where all outcomes in a certain interval are equally likely. When modeling service times or duration of use (such as session length), a uniform distribution assumes that every duration within the given range has an equal chance of occurrence, and the average service time can be calculated as the midpoint of the interval. This simplifies analysis when there is no reason to favor one duration over another.

Queueing Theory

Queueing theory is the mathematical study of waiting lines, or queues, which is used to model systems where 'customers' (such as calls or service requests) compete for 'servers' (such as dial-up lines or service channels). It provides tools for analyzing performance metrics like average number in system, waiting time, and probability of blocking. In systems with a limited number of servers and no waiting space, caused calls are lost or blocked if all servers are busy.

Erlang Loss Model (Erlang B Formula)

The Erlang Loss Model, often calculated using the Erlang B formula, is a key concept in queueing theory for systems with a fixed number of servers and no waiting room. It quantifies the probability that an incoming request will be blocked because all servers are occupied. This model is particularly relevant in telecommunications where it is necessary to determine the optimal number of service channels (or trunks) required to keep the probability of a busy signal (i.e., lost call) below a specified threshold.

Little's Law

Little's Law is a fundamental principle in queueing theory that relates the average number of customers in a system (L) to the average arrival rate (?) and the average time a customer spends in the system (W), through the relation L = ?W. It is a powerful tool for analyzing steady state conditions and helps in understanding system performance, such as the expected number of sessions in progress in a service system.

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