Into a single queue with unlimited buffer size, customers always arrive in pairs but line up one after another in the waiting line for service. One of an arriving pair joins the line ahead of the other, by a random choice. The single server takes up one customer at a time from the head of the queue and serves. Each customer leaves immediately upon completion of his/her service (that is, without waiting for the completion of the partner’s service). Arrivals of pairs are Poisson with a rate of 1 pair per hour. Service times of individual customers are independent and identically distributed Exponential random variables with a mean of 0.25 hour for each customer (and not for a pair). Determine the expected response time of an individual customer.
2. Consider a two-processor server where each processor in enabled with technology that allows the processors to operate at a higher speed under heavy loads, and to operate at a reduced speed under low loads to conserve power. Users submit jobs according to a Poisson process with rate jobs per second. If there is one job in the system, only a single processor is busy, and the processor operates at a speed of 500 MHz (500 million cycles per second). If there are two jobs in the system, both processors are busy, and each operates at a speed of 1000 MHz. If there are three or more jobs in the system, both processors are busy, and each processor operates at a speed of 1500 MHz. The length of a job is modeled as exponentially distributed and requires an average of 500 million CPU cycles. Buffer size is unlimited.
(a) Draw the state diagram for the system, clearly labeling transition rates. (b) Find the steady-state probabilities for the number of jobs in the system. (c) Find the condition for in order for the system to be stable.