In this problem, you are asked to develop a hand $($spreadsheet$)$ simulation for a queuing system and

then to determine several performance metrics for that system. The queuing system consists of a

single queue that feeds two servers. The queue has infinite capacity and FIFO discipline. Server $1$

and Server $2$ each have capacity $1.$ The entity at the top of the queue moves to the first available

server; if both servers are available, the entity moves to Server $1.$

Samples from an arrival process AP are used for entity inter$-$arrival times. Similarly, samples from

two service time processes, SP$1$ and SP$2,$ are used for server service times. The samples from these

random processes are listed in the following sets. You are to assume that each sample value is

generated only when needed in the simulation; the entire random number streams are not visible to

the simulation.

AP $=\{1,1,1,1,3,3,2,5,...\}$

SP$1=\{5,1,2,4,3,...\}$

SP$2=\{6,2,3,5,1,...\}$

The queuing system event set is E $=\{$ A $($arrive at queue$),$ D$1($depart Server $1),$ D$2($depart Server $2),$

T $($Terminate$)\}$ and the state set is S $=\{$ Q$($t$)($number in queue at time t$),$ B$1($t$)($Server $1$ status at

time t; $0=$ idle, $1=$ busy$),$ B$2($t$)($Server $2$ status at time t; $0=$ idle, $1=$ busy$)\}.$ Entities are named by

numbering them by order of arrival and events are named by numbering them by order of occurrence.

The simulation starts empty and idle at t $=0.$ The simulation terminates when simulation time reaches

t $=10.$ When two or more events occur simultaneously, events are processed in the following order:

depart events; terminate event; and arrive events. Simultaneous events of the same type are

processed in increasing order of the entities affected.

Answer the following questions concerning this simulation.

A$.$ Construct an "Event History Table" for this simulation. The rows of this table correspond to

simulation events arranged in order of occurrence. The columns of this table are: $(1)$ Event

Name; $(2)$ Event Time; $(3)$ Event Type; and $(4)$ Entity Affected. $(10$ pts$)$

B$.$ Accurately sketch the system state variables as functions of time. Use separate axes for each

plot, but use the horizontal axis time scale. $(5$ pts$)$

C$.$ Construct an "Entity History Table" for this simulation. The rows of this table correspond to

simulation entities arranged in order of arrival. The columns of this table are: $(1)$ Entity Name;

$(2)$ Arrival Time at Queue; $(3)$ Arrival Time at Server; $(4)$ Departure Time from Server $1$; and

$(5)$ Departure Time from Server $2.(10$ pts$)$

D$.$ Determine the following performance metrics for this system. $(5$ pts$)$

$1.$ Average time in queue.

$2.$ Average number in queue.

$3.$ Average time in system.

$4.$ Average number in system.

$5.$ Server $1$ Utilization

$6.$ Server $2$ Utilization