Queueing Theory

Abstract

Queueing is simply waiting in lines such as stopping at the toll booth, waiting in line for a bank cashier, stopping at a traffic light, waiting to buy stamps at the post office, and so on.

The priest persuades humble people to endure their hard lot, a politician urges them to rebel against it, and a scientist thinks of a method that does away with the hard lot altogether.

—Max Percy

Problems

• 4.1 For the M/M/1 system, find: (a) E(N²), (b) E(N(N − 1)), (c) Var(N).

• 4.2 In an M/M/1 queue, show that the probability that the number of messages waiting in the queue is greater than a certain number m is

  $$ P\left(n>m\right)={\rho}^{m+1} $$

• 4.3 For an M/M/1 model, what effect will doubling λ and μ have on E[N], E[N_q], and E[W]?

• 4.4 Customers arrive at a post office according to a Poisson process with 20 customers/h. There is only one clerk on duty. Customers have exponential distribution of service times with mean of 2 min. (a) What is the average number of customers in the post office? (b) What is the probability that an arriving customer finds the clerk idle?

• 4.5 From the balance equation for the M/M/1 queue, obtain the probability generating function.

• 4.6 An air-line check-in counter at Philadelphia airport can be modeled as an M/M/1 queue. Passengers arrive at the rate of 7.5 customers per hour and the service takes 6 min on the average. (a) Find the probability that there are fewer than four passengers in the system. (b) On the average, how long does each passenger stay in the system? (c) On the average, how many passengers need to wait?

• 4.7 An observation is made of a group of telephone subscribers. During the 2-h observation, 40 calls are made with a total conversation time of 90 min. Calculate the traffic intensity and call arrival rate assuming M/M/1 system.

• 4.8 Customers arrive at a bank at the rate of 1/3 customer per minute. If X denotes the number of customers to arrive in the next 9 min, calculate the probability that: (a) there will be no customers within that period, (b) exactly three customers will arrive in this period, and (c) at least four customers will arrive. Assume this is a Poisson process.

• 4.9 At a telephone booth, the mean duration of phone conversation is 4 min. If no more than 2-min mean waiting time for the phone can be tolerated, what is the mean rate of the incoming traffic that the phone can support?

• 4.10 For an M/M/1 queue operating at fixed ρ = 0.75, answer the following questions: (a) Calculate the probability that an arriving customer finds the queue empty. (b) What is the average number of messages stored? (c) What is the average number of messages in service? (d) Is there a single time at which this average number is in service?

• 4.11 At a certain hotel, a lady serves at a counter and she is the only one on duty. Arrivals to the counter seem to follow the Poisson distribution with mean of 10 customers/h. Each customer is served one at a time and the service time follows an exponential distribution with a mean of 4 min.

  • (a) What is the probability of having a queue?

  • (b) What is the average queue length?

  • (c) What is the average time a customer spends in the system?

  • (d) What is the probability of a customer spending more than 5 min in the queue before being attended to?

    Note that the waiting time distribution for an M/M/1 queue is

    $$ \mathrm{Prob}\left(W>t\right)=W(t)=1-\rho {e}^{-\mu \left(1-\rho \right)t},\kern1em t\ge 0 $$

• 4.12 (a) The probability p_n that an infinite M/M/2 queue is in state n is given by

  $$ {p}_n=\left\{\begin{array}{c}\hfill \frac{\left(1-\rho \right)}{\left(1+\rho \right)},\begin{array}{cc}\hfill \hfill & \hfill n=0\hfill \end{array}\hfill \\ {}\hfill \frac{2\left(1-\rho \right)}{\left(1+\rho \right)}{\rho}^2,\begin{array}{cc}\hfill \hfill & \hfill n\ge 0\hfill \end{array}\hfill \end{array}\right. $$

  where $\rho =\frac{\lambda }{2\mu }$. Find the average occupancy E(N) and the average time delay in the queue E(T).

• 4.13 Consider M/M/k model. Show that the probability of any server is busy is ${\scriptscriptstyle \raisebox{1ex}{$\lambda $}\!\left/ \!\raisebox{-1ex}{$ k\mu $}\right.}$.

• 4.14 For the M/M/1/k system, let q_n be the probability that an arriving customer finds n customers in the system. Prove that

  $$ {q}_n=\frac{p_n}{1-{p}_k} $$

• 4.15 Derive Eq.(4.62) from Eq. (4.61).

• 4.16 Find the mean and variance of the number of customers in the system for the M/M/∞ queue.

• 4.17 At a toll booth, there is only one “bucket” where each driver drops 25 cents. Assuming that cars arrive according to a Poisson probability distribution at rate 2 cars per minute and that each car takes a fixed time 15 s to service, find: (a) the long-run fraction of time that the system is busy, (b) the average waiting time for each car, (c) the average number of waiting cars, (d) how much money is collected in 2 h.

• 4.18 An M/E_k/1 queue has an arrival rate of 8 customers/s and a service rate of 12 customers/s. Assuming that k = 2, find the mean waiting time.

• 4.19 Consider two identical M/M/1 queueing systems in operation side by side in a facility with the same rates λ and μ (ρ = λ/μ). Show that the distribution of the total number N of customers in the two systems combined is

  $$ \mathrm{Prob}\left(N=n\right)=\left(n+1\right){\left(1-\rho \right)}^2{\rho}^n,\kern1em n>0 $$